Color management system based on universal gamut mapping method

ABSTRACT

This invention is a universal gamut mapping and color management method and relates to color digital image processing technical field. It involves high-accuracy image coordinates conversion in device color space, high fidelity chroma transmission between input and output devices, cross-media gamut mapping technologies and color management system built based on these technologies. 
     This invention is accurate, efficient and versatile. It can be widely used in image processing hardware manufacture and software design for image transmission and receiving devices such as computer, digital TV, digital image displays, digital video camera and television camera, etc. Technical features of this method: eliminates red-shift interference, keeps 3 primary colors&#39; independency and channel independency during color synthesis and segmentation, gives priority to grey component reproduction, transfers luminance and chromaticity data in accordance with luminance independent theory, new D l x l y l  profile connection space, accurate color prediction and gamut mapping equation, etc. The combination of all these features guarantees the versatility of this method.

I. TECHNICAL FIELD

This invention is a brand new, practical technology of cross-media imagetransmission and accurate color image reproduction conform to visualeffect. Its main application involve image display devices (e.g. CRT,PDP, LCD or LED monitors), image output devices (e.g. color printers,multicolor offset press, re mote image transmission, image mobilecommunication, network image exchange), associated design and productionfields (e.g. color management system, computer image system, multimediatelevision system, image sending and receiving system). It introduces auniversal and innovative technical solution to software production andhardware manufacture of above system and devices.

II. TECHNICAL BACKGROUND

In current color management system, CIE LAB (CIECAM02) is commonly usedas PCS (profile connection space). However, both color systems CIE LABand CIECAM02 have errors which cannot be neglected. When CMM (colormanagement module) transfer CIE XYZ into RGB or CMYK color space data,device and media have variant attenuation on X, Y, Z values, whichresult in complex non-linear relationship between CIE XYZ and RGB orCMYK. Until now the method to resolve this non-linear relationship isneither unified nor accurate, as it failed to maintain color propertyindependence, channel independence and gray component independenceduring 3 primary color-matching processes. It doesn't take ‘red shift’effect of primary colors into consideration either. As a result, peoplehave to use look-up table method to resolve practical difficulties;look-up table method is complicated and the conversion result is lack ofuniqueness. As to color image reproduction, cross-media gamut mapping ishard to achieve due to lack of universal gamut mapping method. Till nowthe objective of ‘what you see is what you get’ still haven't be metyet. In this technology background, this invention takes a new approachto create a universal cross-media gamut mapping method, which ensures aunified principle, method and mathematical description model can be usedthrough the whole color management system. The new approach establishesthe steps of improving gamut mapping accuracy at the profiling stage toensure the accuracy of color prediction. The use of new gamut mappingmethod and the analytic calculation method not higher than quadraticensures both the gamut mapping accuracy and production efficiency. Thismethod is transparent, has routine procedure and unique result. Itoffers more opportunity for enterprise participation and makes itpossible to maximum the potential use of devices.

III. DESCRIPTION OF THE INVENTION

This invention successfully creates an innovative and unified colorgamut mapping system by solving following major problems:

(1) In process of three primary color reproduction, it is essential tomaintain constant primary property. Currently Marry-Davis formula,Yule-Nielson formula and GOG colorimetric prediction model are notaccurate enough;

(2) In color matching space, 3 primary colors should not only maintaintheir own hue property independent, but also maintain space independencyin each channel; However current color transformation method andmathematical model, such as Neugebauer formula and conventionally colorspace matrix conversion equation, cannot meet these requirements;

(3) Correct gray tone reproduction should be priority in color imagereproduction. This involves gray balance, gray balance curve setting,visual adaption and Gamma correction to media attenuation effect. Withtraditional gamut mapping technology, gray and color reproductioninterface and constraint with each other, therefore cannot ensureprioritized, independent reproduction of gray tone and cause obviouscolor reproduction error in certain gamut area;

(4) So-called uniform color space CIE 1876 L*a*b* and CIE 1976 L*u*v*are not uniform, so it cannot ensure accuracy in gamut mapping and colortransformation; this invention introduces D_(l)x_(l)y_(l) profileconnection space to solve this problem, and detailed process oftransforming D_(l)x_(l)y_(l) will be discussed in following content;

(5) Existing technology cannot resolve the conflict between complexityof transformation model and algorithmic efficiency, so look-up tablemethod is seen as resolution; However look-up table method cannot ensureuniqueness of image mapping and continuity of gamut mapping; asequipment aging and property offset always happen over time, it is hardfor users to take corrective measure;

(6) Existing gamut mapping method is lack of uniform process principleand method to deal with diversity of cross-media color input and outputdevices; existing gamut mapping technology is based on flawed theorywhich incurs random and systematic error;

This invention creates a brand new color mapping method to overcomeabove challenges and ensures color to be mapped keep its original hue,chromaticity coordinates and luminance For easy understanding of theunconventional approaches, first we introduce basic common method usedby various devices in mapping system and related mathematic model. Laterwe will integrate all these fundamental inventions to form a completecolor-mapping system.

Please note, in this document we use unified naming convention and labelsymbol in mathematical models; we only provide description of the mathsymbol when it is used the first time.

1. A color target structure commonly used by input, display and outputdevices with identical driving values.

Objective and usage: this step mainly resolves the unified calibrationproblem among various devices. In order to calibrate the input, displayand output device, it creates a common, connected color targetstructure. Perform the calibration using sample color with the samedriving value and corresponding hue among different devices. Theobjective is to define a unified input standard for different devices incolor management system, and achieve a mapping result with continuityand inheritance features.

Color target's structure and generation steps:

(1) Define the basic input data. The basic input data is the measuredtristimulus value of sample color when creating profile for input,display and output devices. The sample colors on the color target are:

(a) Three primary scale of Monochromatic color: set the driving value to21 levels in the range of 0 to 20, let scale number i=21. For monitor,set the 21 driving values d_(ri), d_(gi), d_(bi) of three primary to:0.00, 12.75, 25.50, 38.25, 51.00, 63.75, 76.50, 89.25, 102.00, 114.75,127.50, 140.25, 153.00, 165.75, 178.50, 191.25, 204.00, 216.75, 229.50,242.25, 255.00, display and measure each color produced by the drivingvalue on monitor. For calibration color target like digital camera ortelevision camera, create photographic paper color target using samedata set.

For printer or scanner color target, set the 21 driving values of threeprimary colors (cmy) d_(ci), d_(mi), d_(yi) to 0%=0.00/255,5%=12.75/255, 10%=25.50/255, 15%=38.25/255, 20%=51.00/255,25%=63.75/255, 30%=76.50/255, 35%=89.25/255, 40%=102.00/255,45%=114.75/255, 50%=127.50/255, 55%=140.25/255, 60%=153.00/255,65%=165.75/255, 70%=178.50/255, 75%=191.25/255, 80%=204.00/255,85%=216.75/255, 90%=229.50/255, 95%=242.25/255, 100%=255/255;

For printer color target, a 21 level monochromatic blank ink scale needsto be added, the driven variable is d_(xi).

(b) Secondary color sample: for monitor, they are three secondary colorproduced by three sets of driving value (d_(r)255+d_(g)255),(d_(r)255+d_(b)255), (d_(g)255+d_(b)255). For printer and scanner, theyare three secondary color produced by (d_(c)100%+d_(m)100%),(d_(m)100%+d_(y)100%), (d_(c)100%+d_(y)100%). For CMYK 4-primaryprinter, the following three secondary color produced by

-   -   (d_(c)100%+d_(k)100%), (d_(m)100%+d_(k)100%),        (d_(y)100%+d_(k)100%) needs to be added;

(c) Tertiary color sample: for monitor, they are gray sample sets withsame driving value (d_(ri)+d_(gi)+d_(ti)) and display sequentially withparameter value ranging from 0 to 255 following color additive process.For printer and scanner, they are gray sample sets with the same drivingvalue (d_(ci)+d_(mi)+d_(yi)), parameter value ranging from 0 to 100%.Although the color target structure of printer and scanner are the same,the material to produce the scanner target and printer target are notthe same and tristimulus of sample color are not identical. Whencreating the profile using color target, the tristimulus values ofsample color need to be measured. Scanner's calibration color target canbe simulated using photographic paper. For CMYK four color printer,three tertiary color produced by (d_(c)100%+d_(m)100%+d_(k)100%),(d_(m)100%+d_(y)100%+d_(k)100%), (d_(c)100%+d_(y)100%+d_(k)100%) and amixed color produced by (d_(c)100%+d_(m)100%+d_(y)100%+d_(k)100%) needto be added as well. Please notes letter d represents the scale drivingvariable parameter. Because CMYK is commonly used to represent the inputvalue of printer device and RGB are commonly used to represent drivingvalue of display and scanner device, we use C, M, Y, K, R, G, B torepresent the calculated data value of variable d_(ci), d_(mi), d_(yi),d_(ki), d_(ri), d_(gi), d_(bi). In order to calculate the CIEXYZ valueof the scanned color later on, not only the CIEXYZ value of the samplecolor needs to be measured using spectrophotometer, but also the RGBaverage value of every sample color needs to be captured with the helpof software.

2. A unified method to keep primary color channel independence forinput, display and output devices

Objective: as a key process in the new gamut mapping method, it ensuresevery primary color component that involves in color-matching hasconstant hue. The primary color channel independency is essential tocolor mixing and gamut mapping. Existing technologies, e.g. Marry-Davisformula, Yule-Nielson formula, Gain-Offset-Gamma formula can't guaranteethe primary color involves in the color matching has constant hue. Forexample, when calculating the dot area using Marry-Davis formula andYule-Nielson formula, the calculated dot area result is a fuzzy valuebetween geometry gain and optical gain. The objective of this method isto resolve this problem.

Method: when driving value varies between 0-255, let all the primary andunit primary have the same hue. If use mathematical way to explain thismethod, this invention uses Liu's primary clamping equation and thederived Liu's primary formula to clamp the primary hue, colorfulness andluminance and ensure independence of primary's hue.

3 steps to keep primary channel independence:

Subtractive primary clamping equation and reference primary formula areused as an example.

Step 1, Measure the tristimulus value of printing sample color onprimary color target using spectrophotometer, given the measured valueas XYZ, measure the tristimulus value of white dot X_(w), Y_(w), Z_(w)and primary solid tristimulus value X_(s), Y_(s), Z_(s). Given theprimary solid tristimulus value X_(s), Y_(s), Z_(s) as the unit primaryvalue of the primary, the following Liu's primary clamping equationrepresents the relationship between clamping primary value a_(t),clamping luminance Y_(t) and color appearance keeping parameter λ:

$\quad\left\{ \begin{matrix}{{\lambda \; X} = {{\left( {1 - a_{t}} \right)X_{w}} + {a_{t}X_{s}}}} \\{{\lambda \; Y_{t}} = {{\left( {1 - a_{t}} \right)Y_{w}} + {a_{t}Y_{s}}}} \\{{\lambda \; Z} = {{\left( {1 - a_{t}} \right)Z_{w}} + {a_{t}Z_{s}}}}\end{matrix} \right.$

This clamping equation is applicable to printer color target, scannercolor target and normally-white-monitor color target. It can also becalled as printer primary clamping equation, scanner primary clampingequation and normally-white-monitor primary clamping equation.

Step 2, solve above Liu's primary clamping equation, and get samplecolor's clamping luminance Y_(t); analytical expression of Y_(t) is asfollows, with which we can calculate sample color's clamping luminanceY_(t):

$Y_{t} = \frac{\left( {Y_{w} - Y_{s}} \right)\begin{Bmatrix}{{Z\left\lbrack {{X_{w}\left( {Y_{w} - Y_{s}} \right)} - {Y_{w}\left( {X_{w} - X_{s}} \right)}} \right\rbrack} -} \\{X\left\lbrack {{Z_{w}\left( {Y_{w} - Y_{s}} \right)} - {Y_{w}\left( {Z_{w} - Z_{s}} \right)}} \right\rbrack}\end{Bmatrix}}{\begin{matrix}{{\left( {Z_{w} - Z_{s}} \right)\left\lbrack {{X_{w}\left( {Y_{w} - Y_{s}} \right)} - {Y_{w}\left( {X_{w} - X_{s}} \right)}} \right\rbrack} -} \\{\left( {X_{w} - X_{s}} \right)\left\lbrack {{Z_{w}\left( {Y_{w} - Y_{s}} \right)} - {Y_{w}\left( {Z_{w} - Z_{s}} \right)}} \right\rbrack}\end{matrix}}$

Step 3, Take clamping luminance Y_(t) into Liu's reference primary valueequation, and get sample color's reference primary value a;

Liu's reference primary value equation is:

$a = \frac{Y_{w} - Y_{t}}{Y_{w} - Y_{s}}$

As shown in above formula, if we let unit primary value=1, then primaryvalue (also called dot area in printing industry) varies between 0 1.When applying Liu's primary clamping equation and Liu's referenceprimary value equation in various occasions, user should discernspecific meaning of parameters in formula. For instance, for printer,primary value a stands for primary value of primary c, m, y (cyan,magenta, yellow); for monitor, a stands for values of primary r, g, b(red, green, blue):

$\begin{matrix}{{c = \frac{Y_{w} - Y_{t}}{Y_{w} - Y_{c}}},} & {{m = \frac{Y_{w} - Y_{t}}{Y_{w} - Y_{m}}},} & {{y = \frac{Y_{w} - Y_{t}}{Y_{w} - Y_{y}}},}\end{matrix}$ $\begin{matrix}\begin{matrix}{{r = \frac{Y_{w} - Y_{t}}{Y_{w} - Y_{r}}},} & {{g = \frac{Y_{w} - Y_{t}}{Y_{w} - Y_{g}}},}\end{matrix} & {b = \frac{Y_{w} - Y_{t}}{Y_{w} - Y_{b}}}\end{matrix}$

c, m, y is so-called dot area in printing industry; if performingcalibration calculation on normally-white LCD, LED monitor, thereference primary value ‘a’ represents the primary value of red r, greeng or blue b etc. The normally-black primary clamping equation andprimary value formula for TV monitor can be reference in specificationof patent PCT/2011/000327.

When creating the profile for scanner, not only the tristimulus valueX_(i), Y_(i), Z_(i) of the 3 primary color scale on the scanner colortarget needs to be measured, the sample color's tristimulus value R_(i),G_(i), B_(i) on the scale also needs to be collected using the datacollection software module. Because X_(i), Y_(i), Z_(i) and R_(i),G_(i), B_(i) actually describe the same sample color in different colorspace, Liu's clamping equation can be slightly modified to substitutethe character X,Y,Z with R,B,G. The updated equation is as follows:

$\quad\left\{ \begin{matrix}{{\lambda \; R} = {{\left( {1 - a_{t}} \right)R_{w}} + {a_{t}R_{s}}}} \\{{\lambda \; G_{t}} = {{\left( {1 - a_{t}} \right)G_{w}} + {a_{t}G_{s}}}} \\{{\lambda \; B} = {{\left( {1 - a_{t}} \right)B_{w}} + {a_{t}B_{s}}}}\end{matrix} \right.$

Apparently this is a characteristic property of scanner device. For thepurpose of distinguishing, it is named as RGB scanner clamping equationin this invention, and the former is named as XYZ scanner clampingequation, which is used when performing RGB-CMY color spacetransformation on scanned sample color. The clamping luminance G_(t) andreference primary value a can be calculated with the RGB scannerclamping equation. The formula is as follows:

$G_{t} = \frac{\left( {G_{w} - G_{s}} \right)\begin{Bmatrix}{{B\left\lbrack {{R_{w}\left( {G_{w} - G_{s}} \right)} - {G_{w}\left( {R_{w} - R_{s}} \right)}} \right\rbrack} -} \\{R\left\lbrack {{B_{w}\left( {G_{w} - G_{s}} \right)} - {G_{w}\left( {B_{w} - B_{s}} \right)}} \right\rbrack}\end{Bmatrix}}{\begin{matrix}{{\left( {B_{w} - B_{s}} \right)\left\lbrack {{R_{w}\left( {G_{w} - G_{s}} \right)} - {G_{w}\left( {R_{w} - R_{s}} \right)}} \right\rbrack} -} \\{\left( {R_{w} - R_{s}} \right)\left\lbrack {{B_{w}\left( {G_{w} - G_{s}} \right)} - {G_{w}\left( {B_{w} - B_{s}} \right)}} \right\rbrack}\end{matrix}}$ $a = \frac{G_{w} - G_{t}}{G_{w} - G_{s}}$

Benefit of the invention: clamping luminance Y_(t) derived from primaryclamping equation suppresses interference from ‘red shift’ whichreducing accuracy of calculated primary values. Primary value a obtainedwith Liu's primary value equation rules out such interference, thereforecolor represented by reference primary value a has same hue as unitprimary value, and tristimulus value of this primary value is no longerequal to measured tristimulus value X, Y, Z, they are modified to X,Y_(t), Z. In following content, reference primary value will be used asindependent parameter in color-matching or 3-primary color-matchingequation, and it provides a precise solution for normalized, systematicand accurate 3-primary color-matching method.

3. The method to keep primary' independence in three channels ofcolor-matching space and Liu's color-matching equation

Method: amongst current technologies, Neugebauer and Masking formula aretypical methods to perform 3-primary color-matching. However theindependency of the primary parameter is poor in three channels due tothe crosstalk between channels. The calculation error could be greaterthan 10.

From 1937 till now, many improvement methods have been proposed, but allfailed to achieve expected accuracy. The popular Look-up table (LUT)algorithm based on interpolation approximation algorithm is a simulationalgorithm which cannot provide accurate color gamut mapping result. Thisinvention set a ‘channel primary value’ parameter for eachcolor-matching channel. The channel primary value can be furthertransformed into the function of reference primary color valueparameter. Meantime, conversion between primary value parameter anddriving parameter to produce the primary color is accurate andreversible. The equation built based on this method can ensure 3 primaryplays independent roles during color-matching in color-matching space.

Types of Liu's color-matching equation:

(1) Liu's subtractive color-matching equation based on subtractive colorreproduction:

In this invention, there are two uses of Liu's subtractivecolor-matching equation:

Firstly, it can be used to create gray balance equation for primary cmyin Liu's four color-matching equations; secondly, it can be used ascolor prediction equation for scanners. For the purpose ofdistinguishing, Liu's subtractive color-matching equation is called3-primary printing color-matching equation in the first use and scanningcolor prediction equation in the second use. Liu's subtractivecolor-matching equation is as follows:

$\quad\left\{ \begin{matrix}{X = \begin{matrix}{{\left( {1 - y_{x\;}} \right)\left( {1 - m_{x}} \right)\left( {1 - c_{x}} \right)X_{w}} + {\left( {1 - y_{x}} \right)\left( {1 - m_{x}} \right)c_{x}X_{c}} + {\left( {1 - y_{x}} \right){m_{x}\left( {1 - c_{x}} \right)}X_{m}} + {{y_{x}\left( {1 - m_{x}} \right)}\left( {1 - c_{x}} \right)X_{y}} +} \\{{y_{x}{m_{x}\left( {1 - c_{x}} \right)}X_{r}} + {{y_{x}\left( {1 - m_{x}} \right)}c_{x}X_{g}} + {\left( {1 - y_{x}} \right)m_{x}c_{x}X_{b}} + {y_{x}m_{x}c_{x}X_{s}}}\end{matrix}} \\{Y = \begin{matrix}{{\left( {1 - y_{y\;}} \right)\left( {1 - m_{y}} \right)\left( {1 - c_{y}} \right)Y_{w}} + {\left( {1 - y_{y}} \right)\left( {1 - m_{y}} \right)c_{y}Y_{c}} + {\left( {1 - y_{y}} \right){m_{y}\left( {1 - c_{y}} \right)}Y_{m}} + {{y_{y}\left( {1 - m_{y}} \right)}\left( {1 - c_{y}} \right)Y_{y}} +} \\{{y_{y}{m_{y}\left( {1 - c_{y}} \right)}Y_{r}} + {{y_{y}\left( {1 - m_{y}} \right)}c_{y}Y_{g}} + {\left( {1 - y_{y}} \right)m_{y}c_{y}X_{b}} + {y_{y}m_{y}c_{y}Y_{s}}}\end{matrix}} \\{Z = \begin{matrix}{{\left( {1 - y_{z\;}} \right)\left( {1 - m_{z}} \right)\left( {1 - c_{z}} \right)Z_{w}} + {\left( {1 - y_{z}} \right)\left( {1 - m_{z}} \right)c_{z}Z_{c}} + {\left( {1 - y_{z}} \right){m_{z}\left( {1 - c_{z}} \right)}Z_{m}} + {{y_{z}\left( {1 - m_{z}} \right)}\left( {1 - c_{z}} \right)Z_{y}} +} \\{{y_{z}{m_{z}\left( {1 - c_{z}} \right)}Z_{r}} + {{y_{z}\left( {1 - m_{z}} \right)}c_{z}Z_{g}} + {\left( {1 - y_{z}} \right)m_{z}c_{z}Z_{b}} + {y_{z}m_{z}c_{z}Z_{s}}}\end{matrix}}\end{matrix} \right.$

In above equation, XYZ on left side represents tristimulus value of thecolor to be matched. The tristimulus X_(w)Y_(w)Z_(w), X_(c)Y_(c)Z_(c),X_(m)Y_(m)Z_(m), X_(y)Y_(y)Z_(y), X_(r)Y_(r)Z_(r), X_(g)Y_(g)Z_(g),X_(b)Y_(b)Z_(b), X_(s)Y_(s)Z_(s) represent 8 solid colors (white, cyan,magenta, yellow, red, green, blue, black mixed with 3 primary) measuredon color target respectively. Please note when dealing with scanner,tristimulus value of 8 solid colors on scanner color target needs to bemeasured; when dealing with printing device, tristimulus value of 8solid colors on color target of printing device needs to be measured;this won't be repeated in following content. Variable y_(x), y_(y),y_(z), m_(x), m_(y), m_(z), c_(x), c_(y), c_(z) represent channelprimary value of yellow, magenta, cyan.

Channel primary parameter c_(x), c_(y), c_(z) are functions of referenceprimary value c;

m_(x), m_(y), m_(z) are functions of reference primary value m;

y_(x), y_(y), y_(z) are functions of reference primary value y. Theformat of the function is:

c_(x)=c^(γ) ^(xc) , c_(y)=c^(γ) ^(yc) , c_(z)=c^(γ) ^(zc) , m_(x)=m^(γ)^(xm) , m_(y)=m^(γ) ^(ym) , m_(z)=m^(γ) ^(zm) , y_(x)=y^(γ) ^(xy) ,y_(y)=y^(γ) ^(yy) , y_(z)=y^(γ) ^(zy)

Please note primary value parameter c, m and y are functions of drivingparameters d_(d), d_(m), d_(y), the functions are as follows:

c=d_(c) ^(γc), m=d_(m) ^(γm), y=d_(y) ^(γy), the inverse solutions are:

d_(c)=c^(1/γc), d_(m)=m^(1/γm), d_(y)=y^(1/γy)

(2) Liu's 4 primary color-matching equation:

This color-matching equation is an extension of above Liu's subtractivecolor-matching equation. For printer and press machine, cmyk 4-primaryreproduction is standard reproduction method. When k=0 or k is a knownvalue, 4-primary color-matching equation degenerates to the normal Liu'ssubtractive color prediction equation which is back to the standardthree primary reproduction process. Liu's 4-primary equation is asfollows:

$\quad\left\{ \begin{matrix}{X = \begin{matrix}{{\left( {1 - c_{x}} \right)\left( {1 - m_{x}} \right)\left( {1 - y_{x}} \right)\left( {1 - k_{dd}} \right)X_{w}} + {{c_{x}\left( {1 - m_{x}} \right)}\left( {1 - y_{x}} \right)\left( {1 - k_{dd}} \right)X_{c}} + {\left( {1 - c_{x}} \right){m_{x}\left( {1 - y_{x}} \right)}\left( {1 - k_{dd}} \right)X_{m}} +} \\{{\left( {1 - c_{x}} \right)\left( {1 - m_{x}} \right){y_{x}\left( {1 - k_{dd}} \right)}X_{y}} + {\left( {1 - c_{x}} \right)m_{x}{y_{x}\left( {1 - k_{dd}} \right)}X_{r}} + {{c_{x}\left( {1 - m_{x}} \right)}{y_{x}\left( {1 - k_{dd}} \right)}X_{g}} +} \\{{c_{x}{m_{x}\left( {1 - y_{x}} \right)}\left( {1 - k_{dd}} \right)X_{b}} + {c_{x}m_{x}{y_{x}\left( {1 - k_{dd}} \right)}X_{s}} + {\left( {1 - c_{x}} \right)\left( {1 - m_{x}} \right)\left( {1 - y_{x}} \right)k_{dd}X_{k}} +} \\{{{c_{x}\left( {1 - m_{x}} \right)}\left( {1 - y_{x}} \right)k_{dd}X_{ck}} + {\left( {1 - c_{x}} \right){m_{x}\left( {1 - y_{x}} \right)}k_{dd}X_{mk}} + {\left( {1 - c_{x}} \right)\left( {1 - m_{x}} \right)y_{k}k_{dd}X_{yk}} +} \\{{\left( {1 - c_{x}} \right)m_{x}y_{x}k_{dd}X_{rk}} + {{c_{x}\left( {1 - m_{x}} \right)}y_{x}k_{dd}X_{gk}} + {c_{x}{m_{x\;}\left( {1 - y_{x}} \right)}k_{dd}X_{bk}} + {c_{x}m_{x}y_{x}k_{dd}X_{sk}}}\end{matrix}} \\{Y = {{\left( {1 - c_{y}} \right)\left( {1 - m_{y}} \right)\left( {1 - y_{y}} \right)\left( {1 - k_{dd}} \right)Y_{w}} + {{c_{y}\left( {1 - m_{y}} \right)}\left( {1 - y_{y}} \right)\left( {1 - k_{dd}} \right)Y_{c}} + \ldots + {c_{y}m_{y}y_{y}k_{dd}Y_{sk}}}} \\{Z = {{\left( {1 - c_{z}} \right)\left( {1 - m_{z}} \right)\left( {1 - y_{z}} \right)\left( {1 - k_{dd}} \right)Z_{w}} + {{c_{z}\left( {1 - m_{z}} \right)}\left( {1 - y_{z}} \right)\left( {1 - k_{dd}} \right)Z_{c}} + \ldots + {c_{z\;}m_{z\;}y_{z}k_{dd}Z_{sk}}}}\end{matrix} \right.$

Variable y_(x), y_(y), y_(z), m_(x), m_(y), m_(z), c_(x), c_(y), c_(z)have the same definition and function as they are in Liu's subtractivecolor-matching equation, which is:

c_(x)=c^(γ) ^(xc) , c_(y)=c^(γ) ^(yc) , c_(z)=c^(γ) ^(zc) , m_(x)=m^(γ)^(xm) , m_(y)=m^(γ) ^(ym) , m_(z)=m^(γ) ^(zm) , y_(x)=y^(γ) ^(xy) ,y_(y)=y^(γ) ^(yy) , y_(z)=y^(γ) ^(zy)

Primary parameter c, m, y are the function of driving parameter d_(d),d_(m), d_(y).

The function is:

c=d_(c) ^(γc), m=d_(m) ^(γm), y=d_(y) ^(γy)

The inverse-solutions are:

d_(c)=c^(1/γ) ^(c) , d_(m)=m^(1/γ) ^(m) , d_(y)=y^(1/γ) ^(y)

The measured value of color target in the equation is same as inprinting 3-primary color-matching equation. k_(dd) represents graycomponent substitute value in black ink. Given k as the referenceprimary of black ink, k_(dd) represents gray component substitute valuein 4-primary reproduction. This will be covered in detail later.Although it appears there are 4 parameters c, m, y, k in the equation,the 4-primary color-matching equation actually is still 3 primaryequation with variable c, m, y because k_(dd) is a known pre-definedvalue. When the equation is used for calibration calculation, theiterative method can be implemented to solve the equation.

(3) Liu's normally-white-monitor color-matching equation(for computer)based on additive color reproduction

In order to create profile for normally-white CRT, PDP, LCD and LEDmonitor and derive the gray balance power function fornormally-white-monitor, the following Liu's normally-white additivecolor-matching equation is used:

$\quad\left\{ \begin{matrix}{X = \begin{matrix}{{\left( {1 - b_{x}} \right)\left( {1 - g_{x}} \right)\left( {1 - r_{x}} \right)X_{w}} + {\left( {1 - b_{x}} \right)\left( {1 - g_{x}} \right)r_{x}X_{r}} + {\left( {1 - b_{x}} \right){g_{x}\left( {1 - r_{x}} \right)}X_{g}} + {{b_{x}\left( {1 - g_{x}} \right)}\left( {1 - r_{x}} \right)X_{b}} +} \\{{b_{x}{g_{x}\left( {1 - r_{x}} \right)}X_{c}} + {{b_{x}\left( {1 - g_{x}} \right)}r_{x}X_{m}} + {\left( {1 - b_{x}} \right)g_{x}r_{x}X_{y}} + {b_{x}g_{x}r_{x}X_{sk}}}\end{matrix}} \\{Y = {{\left( {1 - b_{y}} \right)\left( {1 - g_{y}} \right)\left( {1 - r_{y}} \right)Y_{w}} + {\left( {1 - b_{y}} \right)\left( {1 - g_{y}} \right)r_{y}Y_{r}} + {\left( {1 - b_{y}} \right){g_{y}\left( {1 - r_{y}} \right)}Y_{g}} + \ldots + {b_{y}g_{y}r_{y}Y_{sk}}}} \\{Z = {{\left( {1 - b_{z}} \right)\left( {1 - g_{z}} \right)\left( {1 - r_{z}} \right)Z_{w}} + {\left( {1 - b_{z}} \right)\left( {1 - g_{z}} \right)r_{z}Z_{r}} + {\left( {1 - b_{z}} \right){g_{z}\left( {1 - r_{z}} \right)}Z_{g}} + \ldots + {b_{z}g_{z}r_{z}Z_{sk}}}}\end{matrix} \right.$

In above equation, X, Y, Z represents tristimulus values of color to bematched. X_(w)Y_(w)Z_(w), X_(k)Y_(k)Z_(k) represent measured tristimulusvalue of white dot and black dot on the monitor. X_(r)Y_(r)Z_(r),X_(g)Y_(g)Z_(g), X_(b)Y_(b)Z_(b) represent measured tristimulus value ofprimary color red, green and blue when d_(r), d_(g), d_(b) have theirmaximum values.

X_(c), Y_(c), Z_(c) are the tristimulus value of cyan driven by (G+B)when G and B choose the maximum value;

X_(m), Y_(m), Z_(m) are the tristimulus value of magenta driven by (R+B)when R and B choose the maximum value;

X_(y), Y_(y), Z_(y) are the tristimulus value of yellow driven by (R+G)when R and G choose the maximum value.

The variable parameters r_(x), r_(y), r_(z), g_(x), g_(y), g_(z), b_(x),b_(y), b_(z) on right side of the equation are called channel primaryvalue, they are used to map tristimulus value X, Y, Z on left side ofthe equation. From this point of view, channel primary value has channelindependent property. The channel primary parameter is actually afunction of primary parameter r, g and b instead of a simply variable.

The function is:

r_(x)=r^(γ) ^(xr) , r_(y)=r^(γ) ^(yr) , r_(z)=r^(γ) ^(zr) ; g_(x)=g^(γ)^(xg) , g_(y)=g^(γ) ^(yg) , g_(z)=g^(γ) ^(zg) ; b_(x)=b^(γ) ^(xb) ,b_(y)=b^(γ) ^(yb) , b_(z)=b^(γ) ^(zb)

Please note the primary parameter r, g and b are the function of drivingparameters d_(r), d_(g), d_(b)

The function is as follows:

r=d_(r) ^(γ) ^(rd) , g=d_(g) ^(γ) ^(gd) , b=d_(b) ^(γ) ^(bd) the inversesolution is: d_(r)=r^(1/γ) ^(rd) , d_(g)=r^(1/γ) ^(gd) , d_(b)=r^(1/γ)^(bd)

(4) Liu's normally-black-monitor color-matching equation based onadditive color reproduction: In order to create profile fornormally-blank-monitor and create the gray balance power function fornormally-black TV monitor, the following Liu's normally-black additivecolor-matching equation based on additive method reproduction is used:

$\quad\left\{ \begin{matrix}{X = \begin{matrix}{{\left( {1 - r_{x}} \right)\left( {1 - g_{x}} \right)\left( {1 - b_{x}} \right)X_{k}} + {{r_{x}\left( {1 - g_{x}} \right)}\left( {1 - b_{x}} \right)X_{r}} + {\left( {1 - r_{x}} \right){g_{x}\left( {1 - b_{x}} \right)}X_{g}} + {\left( {1 - r_{x}} \right)\left( {1 - g_{x}} \right)b_{x}X_{b}} +} \\{{\left( {1 - r_{x}} \right)g_{x}b_{x}X_{c}} + {{r_{x}\left( {1 - g_{x}} \right)}b_{x}X_{m}} + {r_{x}{g_{x}\left( {1 - b_{x}} \right)}X_{y}} + {r_{x}g_{x}b_{x}X_{sw}}}\end{matrix}} \\{Y = \begin{matrix}{{\left( {1 - r_{y}} \right)\left( {1 - g_{y}} \right)\left( {1 - b_{y}} \right)Y_{k}} + {{r_{y}\left( {1 - g_{y}} \right)}\left( {1 - b_{y}} \right)Y_{r}} + {\left( {1 - r_{y}} \right){g_{y}\left( {1 - b_{y}} \right)}Y_{g}} + {\left( {1 - r_{y}} \right)\left( {1 - g_{y}} \right)b_{y}Y_{b}} +} \\{{\left( {1 - r_{y}} \right)g_{y}b_{y}Y_{c}} + {{r_{y}\left( {1 - g_{y}} \right)}b_{y}Y_{m}} + {r_{y}{g_{y}\left( {1 - b_{y}} \right)}Y_{y}} + {r_{y}g_{y}b_{y}Y_{sw}}}\end{matrix}} \\{Z = \begin{matrix}{{\left( {1 - r_{z}} \right)\left( {1 - g_{z}} \right)\left( {1 - b_{z}} \right)Z_{k}} + {{r_{z}\left( {1 - g_{z}} \right)}\left( {1 - b_{z}} \right)Z_{r}} + {\left( {1 - r_{z}} \right){g_{z}\left( {1 - b_{z}} \right)}Z_{g}} + {\left( {1 - r_{z}} \right)\left( {1 - g_{z}} \right)b_{z}Z_{b}} +} \\{{\left( {1 - r_{z}} \right)g_{z}b_{z}Z_{c}} + {{r_{z}\left( {1 - g_{z}} \right)}b_{z}Z_{m}} + {r_{z}{g_{z}\left( {1 - b_{z}} \right)}Z_{y}} + {r_{z}g_{z}b_{z}Z_{sw}}}\end{matrix}}\end{matrix} \right.$

The normally-black-monitor color-matching equation in that invention isequivalent to the equation called gray calibration equation in patentPCT/2011/000327.

(5) Liu's RGB scan color-segmentation equation:

In order to create profile for scanner, we need to perform calibrationcalculation for scanner's gray balance function in RGB color space byusing the following RGB scan color-segmentation equation; based on theRGB tristimulus value captured by CCD, RGB scan color-segmentationequation calculates the 3 primary' value (cyan, magenta, yellow) ofscanned scale. In RGB scan color-segmentation equation, c′, m′, y′ areused to represent the primary parameter of color cyan, magenta andyellow. y_(x)′, y_(y)′, y_(z)′, m_(x)′, m_(y)′, m_(z)′, c_(x)′, c_(y)′,c_(z)′ are used to represent channel primary values. In fact, differencebetween XYZ Liu's color-matching equation based on color subtractionmethod and RGB scan color-segmentation equation is similar to thedifference between using kg or lb to weight same object. The equationuses different coordinate system to describe 3 primary colors (cyan,magenta, yellow) in RGB and XYZ color space. In order to differentiateprimary values calculated in different color space, the primary value c,m, y calculated in XYZ color space is called scan primary value; theprimary value c′, m′, y′ calculated in RGB color space is called twinscan primary value. Below is Liu's RGB scanning color-segmentationequation:

$\quad\left\{ \begin{matrix}{R = \begin{matrix}{{\left( {1 - y_{x}^{\prime}} \right)\left( {1 - m_{x}^{\prime}} \right)\left( {1 - c_{x}^{\prime}} \right)R_{w}} + {\left( {1 - y_{x}^{\prime}} \right)\left( {1 - m_{x}^{\prime}} \right)c_{x}^{\prime}R_{c}} + {\left( {1 - y_{x}^{\prime}} \right){m_{x}^{\prime}\left( {1 - c_{x}^{\prime}} \right)}R_{m}} + {{y_{x}^{\prime}\left( {1 - m_{x}^{\prime}} \right)}\left( {1 - c_{x}^{\prime}} \right)R_{y}} +} \\{{y_{x}^{\prime}{m_{x}^{\prime}\left( {1 - c_{x}^{\prime}} \right)}R_{r}} + {{y_{x}^{\prime}\left( {1 - m_{x}^{\prime}} \right)}c_{x}^{\prime}R_{g}} + {\left( {1 - y_{x}^{\prime}} \right)m_{x}^{\prime}c_{x}^{\prime}R_{b}} + {y_{x}^{\prime}m_{x}^{\prime}c_{x}^{\prime}R_{s}}}\end{matrix}} \\{G = \begin{matrix}{{\left( {1 - y_{x}^{\prime}} \right)\left( {1 - m_{x}^{\prime}} \right)\left( {1 - c_{x}^{\prime}} \right)G_{w}} + {\left( {1 - y_{x}^{\prime}} \right)\left( {1 - m_{x}^{\prime}} \right)c_{x}^{\prime}G_{c}} + {\left( {1 - y_{x}^{\prime}} \right){m_{x}^{\prime}\left( {1 - c_{x}^{\prime}} \right)}G_{m}} + {{y_{x}^{\prime}\left( {1 - m_{x}^{\prime}} \right)}\left( {1 - c_{x}^{\prime}} \right)G_{y}} +} \\{{y_{x}^{\prime}{m_{x}^{\prime}\left( {1 - c_{x}^{\prime}} \right)}G_{r}} + {{y_{x}^{\prime}\left( {1 - m_{x}^{\prime}} \right)}c_{x}^{\prime}G_{g}} + {\left( {1 - y_{x}^{\prime}} \right)m_{x}^{\prime}c_{x}^{\prime}G_{b}} + {y_{x}^{\prime}m_{x}^{\prime}c_{x}^{\prime}G_{s}}}\end{matrix}} \\{B = \begin{matrix}{{\left( {1 - y_{x}^{\prime}} \right)\left( {1 - m_{x}^{\prime}} \right)\left( {1 - c_{x}^{\prime}} \right)B_{w}} + {\left( {1 - y_{x}^{\prime}} \right)\left( {1 - m_{x}^{\prime}} \right)c_{x}^{\prime}B_{c}} + {\left( {1 - y_{x}^{\prime}} \right){m_{x}^{\prime}\left( {1 - c_{x}^{\prime}} \right)}B_{m}} + {{y_{x}^{\prime}\left( {1 - m_{x}^{\prime}} \right)}\left( {1 - c_{x}^{\prime}} \right)B_{y}} +} \\{{y_{x}^{\prime}{m_{x}^{\prime}\left( {1 - c_{x}^{\prime}} \right)}B_{r}} + {{y_{x}^{\prime}\left( {1 - m_{x}^{\prime}} \right)}c_{x}^{\prime}B_{g}} + {\left( {1 - y_{x}^{\prime}} \right)m_{x}^{\prime}c_{x}^{\prime}B_{b}} + {y_{x}^{\prime}m_{x}^{\prime}c_{x}^{\prime}B_{s}}}\end{matrix}}\end{matrix} \right.$

Channel primary value in RGB scanning color-segmentation equation isfunction of reference primary value, and its format is same as XYZscanning color-matching equation:

c_(x)′=c′^(γ) ^(xc′) , c_(y)′=c′^(γ) ^(yc′) , c_(z)′=c′^(γ) ^(zc′) ,m_(x)′=m′^(γ) ^(xm′) , m_(y)′=m′^(γ) ^(ym′) , m_(z)′=m′^(γ) ^(zm′) ,y_(x)′=y′^(γ) ^(xy′) , y_(y)′=y′^(γ) ^(yy′) , y_(z)′=y′^(γ) ^(zy′)

The importance of Liu's color-matching equation in this gamut mappingmethod:

When creating the profile for scanner, printer or monitor to achieve thegray balance target, Liu's color-matching equation is used to performcharacteristic calibration for the related gray balance function andobtain the coefficient value in gray balance function which will bereferenced in the profile. Because iteration method is used to solve theLiu's color-matching equation and the result is accurate; it is verysuitable to perform calibration calculation on gray balance polynomialor power function using Liu's color-matching equation. It also makes itpossible to quickly perform color gamut mapping calculation using Liu'scolor gamut mapping equation.

4. A method to characterizing Liu's color-matching equation

Objective of characterization:

After observing the channel primary function in Liu's color-matchingequation, we found channel primary value is the power function ofreference primary value, the constant coefficient in polynomial can becalculated with characterization process;

Procedure: for the 4 types of Liu's color-matching equations mentionedabove, the characterization procedure is the same:

(1) In order to obtain the tristimulus value of three primary scales,the XYZ and RGB tristimulus value of sample color on the color gamutscale needs to be measured. If subscript o, p, q_(w), q_(k) are used todifferentiate the related reference data of scanner, printer,normally-white computer monitor and normally-black TV monitor. the 15tristimulus value arrays can be represented as follows:

-   -   2 pairs of scanner tristimulus value: [X_(oci), Y_(oci),        Z_(oci)], [X_(omi), Y_(omi), Z_(omi)], [X_(oyi), Y_(oyi),        Z_(oyi)] and [R_(oci), G_(oci), B_(oci)], [R_(omi), G_(omi),        B_(omi)], [R_(oyi), G_(oyi), B_(oyi)]    -   printer's tristimulus value: [X_(pci), Y_(pci), Z_(pci)],        [X_(pmi), Y_(pmi), Z_(pmi)], [X_(pyi), Y_(pyi), Z_(pyi)]    -   tristimulus value of normally-white-monitor (computer):        [X_(qwri), Y_(qwri), Z_(qwri)], [X_(qwgi), Y_(qwgi), Z_(qwgi)],        [X_(qwbi), Y_(qwbi), Z_(qwbi)]    -   tristimulus value for normally-black-monitor (television):        [X_(qkri), Y_(qkri), Z_(qkri)], [X_(qkgi), Y_(qkgi), Z_(qkgi)],        [X_(qkbi), Y_(qkbi), Z_(qkbi)]

(2) Calculate the clamping luminance value [Y_(toci), Y_(tomi),Y_(toyi)], [G_(toci), G_(tomi), G_(toyi)], [Y_(tpci), Y_(tpmi),Y_(tpyi)], [Y_(tqwri), Y_(tqwgi), Y_(tqwbi)], [Y_(tqkri), Y_(tqkgi),Y_(tqkbi)] for scanner, printer, normally-white computer monitor andnormally-black TV monitor using primary clamping luminance model basedon the above 15 sets of tristitulus arrays;

(3) Plug clamping luminance value into reference primary formula tocalculate reference primary value:

$\mspace{20mu} \begin{matrix}{{c_{oi} = \frac{Y_{w} - Y_{toci}}{Y_{w} - Y_{c}}},} & {{m_{oi} = \frac{Y_{w} - Y_{tomi}}{Y_{w} - Y_{m}}},} & {y_{oi} = \frac{Y_{w} - Y_{toyi}}{Y_{w} - Y_{y}}}\end{matrix}$ $\mspace{20mu} \begin{matrix}{{c_{oi}^{\prime} = \frac{G_{w} - G_{toci}}{G_{w} - G_{c}}},} & {{m_{oi}^{\prime} = \frac{G_{w} - G_{tomi}}{G_{w} - G_{m}}},} & {y_{oi}^{\prime} = \frac{G_{w} - G_{toyi}}{G_{w} - G_{y}}}\end{matrix}$ $\begin{matrix}\begin{matrix}{{c_{pi} = \frac{Y_{w} - Y_{tpci}}{Y_{w} - Y_{c}}},} & {{m_{pi} = \frac{Y_{w} - Y_{tpmi}}{Y_{w} - Y_{m}}},} & {{y_{pi} = \frac{Y_{w} - Y_{tpyi}}{Y_{w} - Y_{y}}},}\end{matrix} & {k_{pi} = \frac{Y_{w} - Y_{tki}}{Y_{w} - Y_{t}}}\end{matrix}$ $\mspace{20mu} \begin{matrix}{{r_{qwi} = \frac{Y_{w} - Y_{tqwri}}{Y_{w} - Y_{r}}},} & {{g_{qwi} = \frac{Y_{w} - Y_{tqwgi}}{Y_{w} - Y_{g}}},} & {b_{qwi} = \frac{Y_{w} - Y_{tqwbi}}{Y_{w} - Y_{b}}}\end{matrix}$ $\mspace{20mu} \begin{matrix}{{r_{qki} = \frac{Y_{tqkri} - Y_{k}}{Y_{r} - Y_{k}}},} & {{g_{qki} = \frac{Y_{tqkgi} - Y_{k}}{Y_{g} - Y_{k}}},} & {b_{qki} = \frac{Y_{tqkbi} - Y_{k}}{Y_{b} - Y_{k}}}\end{matrix}$

(4) Use following model to calculate channel primary value for Liu'ssubtractive color-matching equation, Liu's 4-primary color-matchingequation and monitor color-matching equation:

c _(xi) , c _(yi) , c _(zi:) c _(x)=(X _(w) −X)/(X _(w) −X _(c)), c_(y)=(Y _(w) −Y)/(Y _(w) −Y _(c)), c _(z)=(Z _(w) −Z)/(Z _(w) −Z _(c))

c_(x), c_(y), c_(z) represent channel primary value of cyan ink in X, Y,Z channel, with c_(x), c_(y), c_(z) we can get channel primary array[c_(xi), c_(yi), c_(zi)]. As for cyan and yellow, we can simplysubstitute the letter c to letter m or y in the above formularespectively to get the corresponding primary value array [m_(xi),m_(yi), m_(zi)] and [Y_(xi), Y_(yi), y_(zi)];

(5) Use following model to calculate channel primary value c_(xi)′,c_(yi)′, c_(zi)′ in RGB scan color-segmentation equation:

c _(x)′=(R _(w) −R)/(R _(w) −R _(c)), c _(y)′=(G _(w) −G)/(G _(w) −G_(c)), c _(z)′=(B _(w) −B)/(B _(w) −B _(c))

c_(r)′, c_(g)′, c_(b)′ represent the channel primary value of cyan inkin R, G, B channel, with c_(r)′, c_(g)′, c_(b)′ we can get channelprimary array [c_(xi), c_(yi), c_(zi)]. As for cyan and yellow, we cansimply substitute letter c with m or y in above formula respectively toget primary value array [m_(ri)′, m_(gi)′, m_(bi)′] and [y_(ri)′,y_(gi)′, y_(bi)′];

(6) Use following model to calculate channel primary value r_(xj),g_(yj), b_(zj) in normally-white-monitor color-matching equation:

r _(x)=(X _(w) −X)/(X _(w) −X _(r)), r _(y)=(Y _(w) −Y)/(Y _(w) −Y_(r)), r _(z)=(Z _(w) −Z)/(Z _(w) −Z _(r))

r_(x), r_(y), r_(z) represent the channel primary value of red in X, Y,Z channel. As for primary green or blue, we can simply substitute theletter r to letter g or b in the above formula respectively to get theprimary value array [g_(xj), g_(yj), g_(zj)] and [b_(xj), b_(yj),b_(zj)];

(7) Use following model to calculate channel primary value r_(xj),g_(yj), b_(zj) in normally-black-monitor color-matching equation:

r _(x)=(X−X _(k))/(X _(r) −X _(k)), r _(y)=(Y−Y _(k))/(Y _(r) −Y _(k)),r _(z)=(Z−Z _(k))/(Z _(r) −Z _(k))

r_(x), r_(y), r_(z) represent the channel primary value of red in X, Y,Z channel. As for primary green or blue, we can simply substitute theletter r to letter g or b in the above formula respectively to get theprimary value array [g_(xj), g_(yj), g_(zj)] and [b_(xj), b_(yj),b_(zj)];

(8) Use curve fitting method to generate channel primary function forLiu' subtractive color-matching equation which is in accordance withsubtractive color reproduction method and used for printer and scanner:perform curve fitting on primary cyan's reference primary value arrayc_(i) and corresponding channel dot area ratio array c_(xi), c_(yi),c_(zi), and the result is function expression of channel primary value;as to primary cyan, we can get following channel primary value functionmodel:

c_(x)=c^(γ) ^(xc) , c_(y)=c^(γ) ^(yc) , c_(z)=c^(γ) ^(zc)

As for primary magenta, we only need to substitute c with m in aboveequations; as for primary yellow, we only need to substitute c with y inabove equations.

(9) Use curve fitting method to generate channel primary function fornormally-white-monitor color-matching equation: perform curve fitting toprimary red's reference primary value array r_(j) and correspondingchannel primary value array r_(xj), r_(yj), r_(zj), and the result isfunction expression of channel primary value; as to primary red, we canget following channel primary function:

r_(x)=r^(γ) ^(xr) , r_(y)=r^(γ) ^(yr) , r_(z)=r^(γ) ^(zr)

As for primary green's channel primary value function, we only need tosubstitute r with g in above equations; as for primary blue's channelprimary value function, we only need to substitute r with g in aboveequations;

(10) Use curve fitting method to generate channel primary function fordigital camera or television camera color-matching equation: (as togenerate channel primary value function for normally-black-monitor,please refer to patent specification PCT/2011/000327). Perform curvefitting on primary red's reference primary value array r_(j) andcorresponding channel primary value arrays r_(xj), r_(yj), r_(zj)respectively, and the result is function expression of channel primaryvalue; as to primary red, we can have following channel primaryfunction:

r_(x)=r^(γ) ^(xr) , r_(y)=r^(γ) ^(yr) , r_(z)=r^(γ) ^(zr)

As for primary green's channel primary value function, we only need tosubstitute r with g in above equations; as for primary blue's channelprimary value function, we only need to substitute r with g in aboveequations;

The above curve fitting steps work out all coefficients in channelspolynomial and the characterization process is completed.

5. A method to generate pure gray scale for scanner, printer,normally-white-monitor, and normally-black-monitor

Objective: quality of gray tone reproduction is prime quality index ofcolor image reproduction; this invention has follows important measure:generate pure ideal gray scale tristimulus value for scanner, printer ormonitor respectively, and use it as base of image gray componentreproduction. Pure gray scale means the gray tristimulus value of thescale is free of ‘red shift’ component.

Procedure:

(1) Create pure gray scale for scanner, printer and monitor: on thescanner or printer's color target and scale sample color displayed onthe monitor, measure the luminance of compound color to get the initialluminance array [Y_(oai)], [G_(o′ai)], [Y_(pai)], [Y_(qwai)],[Y_(qkai)], they are not pure luminance value.

(2) Convert the initial luminance array [Y_(oai)], [G_(o′ai)],[Y_(pai)], [Y_(qwai)], [Y_(qkai)] to initial density array [D_(oai)],[D_(o′ai)], [D_(pai)], [D_(qwai)], [D_(qkai)] as follows:

-   -   [D_(oai)]=lg(Y_(ow)/Y_(oai)), [D_(o′ai)]=lg(G_(ow)/G_(o′ai)),        [D_(pai)]=lg(Y_(pw)/Y_(pai)), [D_(qwai)]=lg(Y_(wq)/Y_(pwai)),

(3) Normalize the initial density array [D_(oai)], [D_(o′ai)],[D_(pai)], [D_(qwai)], [D_(qkai)] to get normalized density array[D_(obi)], [D_(o′bi)], [D_(pbi)], [D_(qwbi)], [D_(qkbi)] as follows:

-   -   [D_(o′bi)]=[D_(o′ai)]/D_(o′amax), [D_(pbi)]=[D_(pai)]/D_(pamax),        [D_(qwbi)]=[D_(qwai)]/D_(qwamax),        [D_(qkbi)]=[D_(qkai)]/D_(qkamax),

In above, D_(oamax), D_(o′amax), D_(pamax), D_(qwamax), D_(qkamax) arethe maximum values in array [D_(oai)], [D_(o′ai)], [D_(pai)],[D_(qwai)], [D_(qkai)].

(4) Normalize the primary scale's driving value to get the normalizeddriving array [d_(i)]. Letter d represents the parameter of scaledriving value. It represents both the driving parameter CMYK and theinput value of driving parameter RGB.

(5) Use normalized driving array [d_(i)] as independent variable arrayand normalized initial density array [D_(obi)], [D_(o′bi)], [D_(pbi)],[D_(qwbi)], [D_(qkbi)] as dependent variable respectively to performpower function fitting and get the gray scale's normalized initialdensity model as follows:

D_(ob)=d̂γ_(o), D_(o′b)=d̂γ_(o′), D_(pb)=d̂γ_(p), D_(qwb)=d̂γ_(qw),D_(qkb)=d̂γ_(qk)

(6) Denormalize D_(ob), D_(o′b), D_(pb), D_(qb), D_(qb) respectively toget denormalized initial density array D_(o), D_(o′), D_(p), D_(qw),D_(qk) as follows:

[D _(oi) ]=[d _(oi)]̂γ_(o) ×D _(oamax) , [D _(o′) ]=[d _(oi)]̂γ_(o′) ×D_(o′amax) , [D _(pi) ]=[d _(oi)]̂γ_(p) ×D _(pamax) , [D _(qwi) ]=[d_(oi)]̂γ_(qw) ×D _(qwamax) , [D _(qki) ]=[d _(oi)]̂γ_(qk) ×D _(qkamax)

Please note although D_(o), D_(o′), D_(p), D_(qw), D_(qk) are calleddenormolized initial density array, their values don't equal to initialdensity array [D_(oai)], [D_(o′ai)], [D_(pai)], [D_(qwi)], [D_(qki)].The purpose of proforming data fitting as above is to optimize theinitial density array. D_(o), D_(o′), D_(p), D_(qw), D_(qk) are the puregray tone density of scanner, printer, normally-white-monitor andnormally-black-monitor we are looking for. Let gray tone of color targetscale=i, the pure gray density array of scanner, printer, normally-whiteand normally-black-monitor are [D_(oi)], [D_(o′i)], [D_(pi)], [D_(qwi)],[D_(qki)].

(7) Create ideal gray luminance array for scanner, printer,normally-white-monitor and normally-black-monitor [Y_(oi)], [G_(oi)],[Y_(pi)], [Y_(qwi)], [Y_(qki)], i.e.

-   -   let [Y_(oi)]=Y_(wo)/(10̂D_(oi)), [G_(oi)]=G_(wo)/(10̂D_(o′i)),        [Y_(pi)]=Y_(wp)/(10̂D_(pi)), [Y_(qwi)]=Y_(ww)/(10̂D_(qwi)),        [Y_(qki)]=Y_(wk)/(10̂D_(qki))

(8) Calculate chromaticity coordinates of reference white dot forscanner, printer and monitor:

Let the white color's tristimulus value on scanner color target X_(ow),Y_(ow), Z_(ow) be the white dot's tristimulus value in XYZ color space.

Let the white color's tristimulus value on scanner color target R_(ow),G_(ow), B_(ow) be the white dot's tristimulus value in RGB color space.

Let white color's tristimulus value on printer color target X_(pw),Y_(pw), Z_(pw) be the white dot's tristimulus value in XYZ color space.

Let chromaticity coordinates of D65 illuminant be the reference whitedot's tristimulus value on monitor to get R_(ow)G_(ow)B_(ow). Given o,p, q as the sub coordinator of scanner, printer and monitor, thechromaticity coordinates of reference white dot can be calculated asfollows:

x _(ow) =X _(ow)/(X _(ow) +Y _(ow) +Z _(ow)), y _(ow) =Y _(ow)/(X _(ow)+Y _(ow) +Z _(ow))

r _(ow) =R _(ow)/(R _(ow) +G _(ow) +B _(ow)), g _(ow) =G _(ow)/(R _(ow)+G _(ow) +B _(ow))

x _(pw) =X _(pw)/(X _(pw) +Y _(pw) +Z _(pw)), y _(pw) =Y _(pw)/(X _(pw)+Y _(pw) +Z _(pw))

x_(qw)=x_(qk)=0.3127, y_(qw)=x_(qk)=0.3290

(9) Using pure gray luminance array [Y_(oi)], [G_(oi)], [Y_(pi)],[Y_(qwi)], [Y_(qki)] from 7^(th) step above and chromaticity coordinatesx_(ow), y_(ow), r_(ow), g_(ow), x_(pw), y_(pw), x_(qw), y_(qw), x_(qk),y_(qk) to create pure gray scale for scanner, printer,normally-white-monitor and normally-black-monitor respectively; eachpure gray scale's tristimulus value is calculated as follows:

$\begin{matrix}{{{{Scanner}\mspace{14mu} X_{oi}} = {\frac{x_{ow}}{y_{ow}} \cdot Y_{oi}}},} & {{Y_{oi} = Y_{oi}},} & {Z_{oi} = {\frac{\left( {1 - x_{ow} - y_{ow}} \right)}{y_{ow}} \cdot Y_{oi}}}\end{matrix}$ $\begin{matrix}{{R_{oi} = {\left( {r_{ow}/g_{ow}} \right) \cdot G_{oi}}},} & {{G_{oi} = G_{oi}},} & {B_{oi} = {\left( {1 - r_{ow} - g_{ow}} \right) \cdot G_{oi}}}\end{matrix}$ $\begin{matrix}{{{{Printer}\mspace{14mu} X_{pi}} = {\frac{x_{pw}}{y_{pw}} \cdot Y_{pi}}},} & {{Y_{pi} = Y_{pi}},} & {Z_{pi} = {\frac{\left( {1 - x_{pw} - y_{pw}} \right)}{y_{pw}} \cdot Y_{pi}}}\end{matrix}$ $\begin{matrix}{{{{Normally}\text{-}{white}\text{-}{monitor}\mspace{14mu} X_{qwi}} = {\frac{x_{qw}}{y_{qw}} \cdot Y_{qwi}}},} & {{Y_{qwi} = Y_{qwi}},}\end{matrix}$$Z_{qwi} = {\frac{\left( {1 - x_{qw} - y_{qw}} \right)}{y_{qw}} \cdot Y_{qwi}}$$\begin{matrix}{{{{Normally}\text{-}{b{lack}}\text{-}{monitor}\mspace{14mu} X_{qki}} = {\frac{x_{qk}}{y_{qk}} \cdot Y_{qki}}},} & {{Y_{qki} = Y_{qki}},}\end{matrix}$$Z_{qki} = {\frac{\left( {1 - x_{qk} - y_{qk}} \right)}{y_{qk}} \cdot Y_{qki}}$

6. A method to characterize component primary values of pure gray scale

Objective: The objective of using Liu's color-matching equation tocharacterize Liu's pure gray scale is to calculate pure gray scale'scomponent primary value and then present it as function of pure graydensity parameter. Gray balance is a critical index to the quality ofcolor image reproduction. The primary function expression at graybalance status can be used to derive any color's gray core component;gray core can be viewed from following respect: a color is mixed withprimary c, m, y, the amount of these three primary are unequal value,and the primary color with least reference primary value is gray core ofthis color's gray component; gray core is the least value of 3 primary,together with other 2 primary, gray core of a composite color isproduced. With the help of gray core concept, this invention achievedthe objective to segment a color into 3 primary colors rapidly andaccurately, and reproduce gray component with priority.

Procedure:

(1) In scanner's RGB color space, create gray balance power function c′(D_(rgb)), g′ (D_(rgb)), b′ (D_(rgb)) for scanner. The steps are:perform color-matching calculation for scanner's gray scale array[R_(oi),G_(oi),B_(oi)] using RGB scan color-segmentation equation; theresult is scanner gray scale's reference primary value array [c_(i)′,m_(i)′, y_(i)′], then calculate gray scale's density array [D_(rgb)]with gray scale luminance array [G_(oi)]: plug luminance array [G_(oi)]into density formula D_(rgb)=lg(G_(ow)/G_(oi)) and get density array[D_(rgbi)]; finally perform data fitting with [D_(rgb)] as independentvariable and twin primary values [c′_(i), m′_(i), y′_(i)] as dependentvariables respectively, and the result is scanner's gray balance powerfunction as follows:

c_(dd)′=D_(rgb) ^(γ) ^(co′) , m_(dd)′=D_(rgb) ^(γ) ^(mo′) ,y_(dd)′=D_(rgb) ^(γ) ^(yo′)

In following scanner color-segmentation equation, twin primary valuesc_(dd)′, m_(dd)′, y_(dd)′ will be used as gray core.

(2) Create c′m′y′-cmy power function for scanner, with which primaryvalue c′, m′, y′ can be converted to primary value c, m, y; the methodis: perform color-matching calculation for array [X_(oi), Y_(oi),Z_(oi)] with Liu's subtractive color-matching equation, and getscanner's reference primary array [c_(i), m_(i), y_(i)] which forms grayscale in CMY color space, then perform curve fitting with [c_(i)′],[m_(i)′], [y_(i)′] as independent variables and [c_(i)], [m_(i)],[y_(i)] as dependent variable, and result is power function conversionexpression for c′m′y′-cmy:

c=c′^(γ) ^(cc′) , m=m′^(γ) ^(mm′) , y=y′^(γ) ^(yy′)

(3) Create gray balance primary power function for computer'snormally-white-monitor and television's normally-black-monitor: there isonly one slight difference between color-matching equations ofcomputer's normally-white-monitor and television'snormally-black-monitor. We use superscript w and subscript k to denotetwo cases: perform color-matching calculation for gray scale array[X_(qwi), Y_(qwi), Z_(qwi)] and [X_(qki), Y_(qki), Z_(qki)] usingnormally-white-monitor color-matching equation andnormally-black-monitor color-matching equation, and we get gray scale'sprimary value array [r_(qwi), g_(qwi), b_(qwi)], [r_(qki), g_(qki),b_(qki)]; then perform curve fitting with pure gray density arrays[D_(qwi)], [D_(qki)] as independent variables and [r_(qwi), g_(qwi),b_(qwi)], [r_(qki), g_(qki), b_(qki)] as dependent variablesrespectively to get the gray balance primary function fornormally-white-monitor and normally-black-monitor:

r_(qw)=D_(qw) ^(γ) ^(rN) , g_(qw)=D_(qw) ^(γ) ^(gw) , b_(qw)=D_(ow) ^(γ)^(bw)

r_(qk)=D_(qk) ^(γ) ^(rk) , g_(qk)=D_(qk) ^(γ) ^(gk) , b_(qk)=D_(ok) ^(γ)^(bk)

Inverse functions of above are:

D_(qw)=r_(qw) ^(1/γ) ^(rw) , D_(qw)=g_(qw) ^(1/γ) ^(gw) , D_(ow)=b_(qw)^(1/γ) ^(bw)

D_(qk)=r_(qk) ^(1/γ) ^(rk) , D_(qk)=g_(qk) ^(1/γ) ^(gk) , D_(ok)=b_(qk)^(1/γ) ^(bk)

r_(qw), g_(qw), b_(qw), r_(qk), g_(qk), b_(qk) are component primaryvalue calculated based on pure neutral gray color; to emphasize thisspecial property, we call both r_(qw), g_(qw), b_(qw) and r_(qk),g_(qk), b_(qk) gray balance primary values; one of them will be used asgray core of chromatic color. In other words, amongst 3 componentprimary values of ideal gray color, only the component primary used asgray core can be called gray core.

(4) Create gray balance power function for 3 primary color printer:

Perform color-matching calculation for gray scale array [X_(pi), Y_(pi),Z_(pi)] with Liu's 3 primary color-matching equation, and the result isgray scale's primary array [c_(pi), m_(pi), y_(pi)]; then perform curvefitting with pure gray density array [D_(pi)] as independent variableand [c_(pi), m_(pi), y_(pi)] as dependent variable, thus we get powerfunction of 3 primary printer gray balance primary value as follows:

c_(p)=D_(p) ^(γ) ^(cd) , m_(p)=D_(p) ^(γ) ^(md) , y_(p)=D_(p) ^(γ) ^(yd)

Inverse functions of above are:

D_(p)=c_(p) ^(1/γ) ^(cd) , D_(p)=m_(p) ^(1/γ) ^(md) , D_(p)=y_(p) ^(1/γ)^(yd)

Above gray balance power function is called 3 primary printer graybalance power function. c_(p), m_(p), y_(p) is component primary valuebased on pure neutral gray [X_(pi), Y_(pi), Z_(pi)]; c_(p), m_(p), y_(p)calculated with Liu' subtractive color-matching equation is not used as‘gray core’, but for calculating cmy's driving value in Liu's 4 primarymapping equation. In this invention, 4 primary color printer's pureneutral gray is [X_(pi), Y_(pi), Z_(pi)] as well, while componentprimary value of [X_(pi), Y_(pi), Z_(pi)] also includes graysubstitution parameter k_(p).

(5) Create gray balance polynomial for 4-primary color printer (or 4primary color press):

a. Determine gray component substitute value: measure black ink printingscale on color target, record tristimulus value of sample color;calculate black primary's clamping luminance array [Y_(tki)] withclamping luminance equation; then calculate black primary's referenceprimary array k_(i) based on black primary's clamping luminance Y_(tki):

k _(i)=(Y _(wp) −Y _(tki))/(Y _(wp) −Y _(sk))

b. Perform curve fitting with normalized driving array [d_(ki)] asindependent array and [k_(i)] as dependent array and the result is:k=d_(k)̂γ_(k)

c. Gray substitute value is denoted as k_(dd), letk_(dd)=Q(d_(k)̂γ_(k))^(n); Q is a pre-set proportion constant value forcontrolling maximum black print value; n is a index value based on tonelength of black print. n can be used to easily adjust tone of blackprint. Please note, gray substitute value k_(p) is purified blackcomponent and is called pure gray component substitute value;

d. Solve pure gray's primary value array [c_(i)], [m_(i)], [y_(i)]. Asto pure gray scale, there are i sets of tristimulus value [X_(pi),Y_(pi), Z_(pi)]; if each set of tristimulus value is plug in left sideof 4 primary color matching equation one by one, and at same timecorresponding [k_(pi)] is plug into same equation, then 4 primarycolor-matching equation turns into a static Liu's color-matchingequation with only c, m, y as its variables; meanwhile solve equationusing iteration method to get reference primary array [c_(i)], [m_(i)],[y_(i)];

e. Preform curve fitting with [D_(pi)] as independent variable and[c_(pi)], [m_(pi)], [y_(pi)], [k_(pi)] as dependent variables, andresults are gray balance polynomial for 4 primary color printing,primary value c_(p), m_(p), y_(p) and gray component substitute valuek_(p) as follows:

c _(p) =a ₀ +a ₁ D _(p) +a ₂ D _(p) ² +a ₃ D _(p) ³ + . . . , m _(p) =b₀ +b ₁ D _(p) +b ₂ D _(p) ² +b ₃ D _(p) ³+ . . .

y _(p) =c ₀ +c ₁ D _(p) +c ₂ D _(p) ² +c ₃ D _(p) ³ + . . . , k _(p) =d₀ +d ₁ D _(p) +d ₂ D _(p) ² +d ₃ D _(p) ³+ . . .

c_(p), m_(p), y_(p) calculated from above function are component primaryvalues calculated based on visual neutral gray under gray balancecondition; to emphasize this special feature, we call c_(p), m_(p),y_(p) gray balance primary value in printing color space. In printingcolor space, amongst c_(p), m_(p), y_(p), there must be a gray balanceprimary value, together with k_(p), to form gray core which conform tovisual effect, and this gray balance primary value is called gray core.

7. A method to perform Gamma correction for gray tone of image

Objective: because image reproduction media and device tend to dull thetone of display and printing image, we have to do overall Gammacorrection to image tone. Gamma correction method adopted by thisinvention is totally different from conventionally technology andensures Gamma correction and gamut mapping can be carried out at samephase.

Procedure:

(1) Express pure gray density parameter as function of drivingparameter: [D_(pi)], [D_(qwi)], [D_(qki)] are pure gray density arraysof printer, normally-black-monitor and normally-black-monitor mentionedin article 5 section (6); normalize [D_(pi)], [D_(qwi)], [D_(qki)] andthe results are still denoted as [D_(pi)], [D_(qwi)], [D_(qki)]; then weexpress them as function of normalized driving parameter [d_(i)]:perform curve fitting with [d_(i)] as independent variable and [D_(pi)],[D_(qwi)], [D_(qki)] as dependent variables, the result is powerfunction expression of pure gray density:

D_(p)=d^(γ) ^(p) , D_(qw)=d^(γ) ^(qw) , D_(qk)=d^(γ) ^(qk)

(2) Express Gamma correction density as function of pure gray density,i.e. express corrected density D_(p)′, D_(qw)′, D_(qk)′ as function oforiginal density D_(p), D_(qw), D_(qk):

D_(p)′=d_(p) ^(1/γ) ^(p) =D_(p) ^(1/γ) ^(p) ² , D_(qw)′=d_(qw) ^(1/γ)^(qw) =D_(qw) ^(1/γ) ^(qw) ² , D_(qk)′=d_(k) ^(1/γ) ^(qk) =D_(qk) ^(1/γ)^(qk) ²

Above three makes up Liu's Gamma correction equations for Gammacorrection density calculation.

8. A method to create D_(l)x_(l)y_(l) profile connection color space forscanner in XYZ color space

Objective: color input devices such as digital camera and scanner needto transfer captured color information to color output devices such asmonitor and printer. Existing methods use CIE LAB (CIECAM02) as PCSprofile connection space; it is correct to use uniform color space aslink, however, such uniform color space is not ideally uniform;therefore using it as PCS profile connection space leads to obviouscolor conversion error and involves tedious steps. The other problem isrelated to color value transfer method. Cross-media color transmissionover long distance is very common today, mobile communication, digitaltelevision, and ground-to-air image communication broke the pattern ofcolor management limitation on ‘one time one location’. Traditionalmethod to generate luminance signal and color difference signals cannotmeet constant luminance principle in non-lineal condition, which worsensreproduction quality of television image details. To solve this problem,this invention offers a brand new, multi-functional D_(l)x_(l)y_(l)profile connection space; the method to create D_(l)x_(l)y_(l) profileconnection color space in XYZ space is explained as below.

Procedure: D_(l)x_(l)y_(l) color space is created using Liu'scolor-segmentation equation. The equation is based on following theory:a color is generated in accordance with 3 primary color compositiontheory; while from a new respective, we can suppose: a color is alwayscomposed of two components: one is gray component and its percentage isp, the other is secondary color component composed of 2 primary colorsand its percentage is (1−p), so the new structure is [1 visual graycomponent+2 primary components]. Both perspectives are equal. The newone seems more complex in the first place; however such complexitybrings us many benefits, such as turning high order algorithm intosimple analytic algorithm thus enhancing computing efficiency. Liu'scolor-segmentation equation has 2 sub-types: the one for scanner iscalled Liu's scanning color-segmentation equation; the one for digitalcamera, digital video camera and television camera is called Liu'sphotographed color-segmentation equation.

As color obtained by scanner usually is transformed to gamut of printingdevice, Liu's color-segmentation equation based on subtractivecolor-reproduction is chosen;

As color obtained by digital camera usually is transformed tonormally-white-monitor, Liu's color-segmentation equation based onnormally-white-monitor color-matching equation is chosen;

As color obtained by television camera is usually transformed tonormally-black-monitor, Liu's color-segmentation equation based onnormally-black-monitor color-matching equation is chosen; each Liu'scolor-segmentation equation has 3 sub-types.

(1) A method to create D_(l)x_(l)y_(l) profile connection color spacefor scanner in XYZ color space

Step 1, Create Liu's color-segmentation equation for scanner:

Format of Liu's color-segmentation equation for scanner: 3 sub-types areas follows:

$\left\{ {\begin{matrix}{X = {{\begin{bmatrix}{{\left( {1 - c} \right)\left( {1 - m} \right)X_{w}} + {{c\left( {1 - m} \right)}X_{c}} +} \\{{\left( {1 - c} \right)m\; X_{m}} + {cmX}_{b}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot X_{sk}}}} \\{Y = {{\begin{bmatrix}{{\left( {1 - c} \right)\left( {1 - m} \right)Y_{w}} + {{c\left( {1 - m} \right)}Y_{c}} +} \\{{\left( {1 - c} \right)m\; Y_{m}} + {cmY}_{b}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot Y_{sk}}}} \\{Z = {{\begin{bmatrix}{{\left( {1 - c} \right)\left( {1 - m} \right)Z_{w}} + {{c\left( {1 - m} \right)}Z_{c}} +} \\{{\left( {1 - c} \right)m\; Z_{m}} + {cmZ}_{b}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot Z_{sk}}}}\end{matrix}\left\{ {\begin{matrix}{X = {{\begin{bmatrix}{{\left( {1 - c} \right)\left( {1 - y} \right)X_{w}} + {{c\left( {1 - y} \right)}X_{c}} +} \\{{\left( {1 - c} \right)y\; X_{y}} + {cyX}_{g}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot X_{sk}}}} \\{Y = {{\begin{bmatrix}{{\left( {1 - c} \right)\left( {1 - y} \right)Y_{w}} + {{c\left( {1 - y} \right)}Y_{c}} +} \\{{\left( {1 - c} \right)y\; Y_{y}} + {cyY}_{g}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot Y_{sk}}}} \\{Z = {{\begin{bmatrix}{{\left( {1 - c} \right)\left( {1 - y} \right)Z_{w}} + {{c\left( {1 - y} \right)}Z_{c}} +} \\{{\left( {1 - c} \right)y\; Z_{y}} + {cyZ}_{g}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot Z_{sk}}}}\end{matrix}\left\{ \begin{matrix}{X = {{\begin{bmatrix}{{\left( {1 - c} \right)\left( {1 - y} \right)X_{w}} + {{m\left( {1 - y} \right)}X_{m}} +} \\{{\left( {1 - m} \right)y\; X_{y}} + {myX}_{r}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot X_{sk}}}} \\{Y = {{\begin{bmatrix}{{\left( {1 - c} \right)\left( {1 - y} \right)Y_{w}} + {{m\left( {1 - y} \right)}Y_{m}} +} \\{{\left( {1 - m} \right)y\; Y_{y}} + {myY}_{r}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot Y_{sk}}}} \\{Z = {{\begin{bmatrix}{{\left( {1 - c} \right)\left( {1 - y} \right)Z_{w}} + {{m\left( {1 - y} \right)}Z_{m}} +} \\{{\left( {1 - m} \right)y\; Z_{y}} + {myZ}_{r}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot Z_{sk}}}}\end{matrix} \right.} \right.} \right.$

Liu's color-segmentation equation is a set of equations comprising 3basic equations; the difference amongst them is:

One is color-segmentation equation with [1 primary cyan+1 primarymagenta+visual gray]

One is color-segmentation equation with [1 primary cyan+1 primaryyellow+visual gray]

One is color-segmentation equation with [1 primary magenta+1 primaryyellow+visual gray]

Simply put, we can call them color-segmentation equation CMK,color-segmentation equation CYK and color-segmentation equation MYK.Three equations seem complex, but they are actually simply quadraticequations and can be solved rapidly with simple analytic method.

In above equations, X, Y, Z are tristimulus values obtained by scanner,

X_(w), Y_(w), Z_(w) are tristimuls of solid white measured from scannertarget,

X_(c), Y_(c), Z_(c) are tristimuls of solid cyan,

X_(m), Y_(m), Z_(m) are tristimuls of solid magenta,

X_(y), Y_(y), Z_(y) are tristimuls of solid yellow,

X_(sk), Y_(sk), Z_(sk) are tristimuls values of black dot composed ofsolid cyan, solid magenta and solid yellow, and have same chromaticitycoordinates as X_(w), Y_(w), Z_(w).

With help of computer program, we can choose one of threecolor-segmentation equations to perform color-segmentation and get grayvalue p in XYZ.

In Liu's color-segmentation equation, let the value in square bracketson right side equal to sold white color's tristimulus value X_(w),Y_(w), Z_(w) measured on scanner color target, then the equation turnsinto its gray scale format as follows:

X _(v) =X _(w)(1−p)+pX _(sk)

Y _(v) =Y _(w)(1−p)+pY _(sk)

Z _(v) =Z _(w)(1−p)+pZ _(sk)

If we plug [p_(i)] into left side of gray scale format of Liu'scolor-segmentation equation respectively, the resulted tristimulus value[R_(vi), G_(vi), B_(vi)] is equal to tristimulus value [R_(oi), G_(oi),B_(oi)] of pure gray scale.

Step 2, Create D_(l)x_(l)y_(l) profile connection color space usingabove concept:

(a) Calculate chromaticity coordinates of color X_(v)Y_(v)Z_(v) based onvalue X_(v)Y_(v)Z_(v) on left side of Liu's color-segmentation equation:with this set of XYZ, we can calculate x_(l), y_(l) required byD_(l)x_(l)y_(l) profile connection color space as below:

$\begin{matrix}{x_{l} = {x_{o} = \frac{X}{X + Y + Z}}} & {y_{l} = {y_{o} = \frac{Y}{X + Y + Z}}}\end{matrix}$

(b) Segment XYZ using Liu's color-segmentation equation: calculate grayvalue p by solving Liu's color-segmentation equation;

(c) Plug p into luminance formula of Liu's color-segmentation equation(gray scale format), then calculate gray scale's luminance value incolor X_(l)Y_(l)Z_(l):

Y _(l) =Y _(v) =Y _(w)(1−p)+pY _(sk)

(d) Convert gray scale's luminance value Y_(l) into gray density valueD_(l): D_(l)=D_(o)=lg(Y_(wo)/Y_(o))

After above 4 steps, any color XYZ in XYZ color space or scanned colorXYZ has been converted into D_(o)x_(o)y_(o) of profile connection colorspace.

(2) A method to create D_(l)x_(l)y_(l) profile connection color spacefor scanner in RGB color space

Step 1. Create Liu's color-segmentation equation for scanner:

(a) Format of Liu's color-segmentation equation for scanner In RGB colorspace. 3 sub-types are as follows,

$\left\{ {\begin{matrix}{R = {{\begin{bmatrix}{{\left( {1 - c^{\prime}} \right)\left( {1 - m^{\prime}} \right)R_{w}} + {{c\left( {1 - m^{\prime}} \right)}R_{c}} +} \\{{\left( {1 - c^{\prime}} \right)m^{\prime}\; R_{m}} + {c^{\prime}m^{\prime}R_{b}}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot R_{sk}}}} \\{G = {{\begin{bmatrix}{{\left( {1 - c^{\prime}} \right)\left( {1 - m^{\prime}} \right)G_{w}} + {{c\left( {1 - m^{\prime}} \right)}G_{c}} +} \\{{\left( {1 - c^{\prime}} \right)m^{\prime}\; G_{m}} + {c^{\prime}m^{\prime}G_{b}}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot G_{sk}}}} \\{B = {{\begin{bmatrix}{{\left( {1 - c^{\prime}} \right)\left( {1 - m^{\prime}} \right)B_{w}} + {{c\left( {1 - m^{\prime}} \right)}B_{c}} +} \\{{\left( {1 - c^{\prime}} \right){m\;}^{\prime}B_{m}} + {c^{\prime}m^{\prime}B_{b}}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot B_{sk}}}}\end{matrix}\left\{ {\begin{matrix}{R = {{\begin{bmatrix}{{\left( {1 - c^{\prime}} \right)\left( {1 - y^{\prime}} \right)R_{w}} + {{c\left( {1 - y^{\prime}} \right)}R_{c}} +} \\{{\left( {1 - c^{\prime}} \right)y^{\prime}\; R_{y}} + {c^{\prime}y^{\prime}R_{g}}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot R_{sk}}}} \\{G = {{\begin{bmatrix}{{\left( {1 - c^{\prime}} \right)\left( {1 - y^{\prime}} \right)G_{w}} + {{c\left( {1 - y^{\prime}} \right)}G_{c}} +} \\{{\left( {1 - c^{\prime}} \right)y^{\prime}\; G_{y}} + {c^{\prime}y^{\prime}G_{g}}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot G_{sk}}}} \\{Z = {{\begin{bmatrix}{{\left( {1 - c^{\prime}} \right)\left( {1 - y^{\prime}} \right)B_{w}} + {{c\left( {1 - y^{\prime}} \right)}B_{c}} +} \\{{\left( {1 - c^{\prime}} \right){y\;}^{\prime}B_{y}} + {c^{\prime}y^{\prime}B_{g}}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot B_{sk}}}}\end{matrix}\left\{ \begin{matrix}{R = {{\begin{bmatrix}{{\left( {1 - m^{\prime}} \right)\left( {1 - y^{\prime}} \right)R_{w}} + {{m\left( {1 - y^{\prime}} \right)}R_{m}} +} \\{{\left( {1 - m^{\prime}} \right)y^{\prime}\; R_{y}} + {m^{\prime}y^{\prime}R_{r}}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot R_{sk}}}} \\{G = {{\begin{bmatrix}{{\left( {1 - m^{\prime}} \right)\left( {1 - y^{\prime}} \right)G_{w}} + {{m\left( {1 - y^{\prime}} \right)}G_{m}} +} \\{{\left( {1 - m^{\prime}} \right)y^{\prime}\; G_{y}} + {m^{\prime}y^{\prime}G_{r}}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot G_{sk}}}} \\{B = {{\begin{bmatrix}{{\left( {1 - m^{\prime}} \right)\left( {1 - y^{\prime}} \right)B_{w}} + {{m\left( {1 - y^{\prime}} \right)}B_{m}} +} \\{{\left( {1 - m^{\prime}} \right){y\;}^{\prime}B_{y}} + {m^{\prime}y^{\prime}B_{r}}}\end{bmatrix} \cdot \left( {1 - p} \right)} + {p \cdot B_{sk}}}}\end{matrix} \right.} \right.} \right.$

In above equation, let values in square bracket on right side ofequation equals to solid white color's tristimulus value measured onscanner color target, i.e. R_(w), G_(w), B_(w), then Liu'scolor-segmentation equation turns into gray scale format of Liu'scolor-segmentation equation as follows:

R _(v) =R _(w)·(1−p)+p·R _(sk)

G _(v) =G _(w)·(1−p)+p·G _(sk)

V _(v) =B _(w)·(1−p)+p·B _(sk)

If we plug [p_(i)] into left side of each Liu's color-segmentationequation (gray scale format), then the generated tristimulus array[R_(vi), G_(vi), B_(vi)] is equivalent to pure gray scale's tristimulusvalue [R_(oi), G_(oi), B_(oi)];

Step 2, Calculate gray density value for gray component R_(v), G_(v),B_(v):

-   -   (a) Segment RGB using Liu's color-segmentation equation, i.e.        calculate gray value p by solving Liu's color-segmentation        equation;    -   (b) Plug value p into luminance calculation expression in Liu's        color-segmentation equation (gray scale format) to calculate        gray scale's luminance value of R_(v)G_(v)B_(v):

G _(l) =G _(v) =G _(w)(1−p)+pG _(sk)

-   -   (c) Convert gray scale's luminance value G_(l) into gray density        value D_(rgb) as:

D _(rgb) =D _(l) =lg(G _(wo) /G _(v))

Benefit of D_(l)x_(l)y_(l) profile connection color space:

(1) The conversion from XYZ to D_(l)x_(l)y_(l) is accurate and won'tincur error for following mapping transformation;

(2) D_(l)x_(l)y_(l) profile connection color space is created usingLiu's color-segmentation equation as a tool; in essence, the process isseparate a color XYZ into luminance signal D_(l) and chromaticity signalx_(l)y_(l). As to television and satellite communication, with the helpof parameter p in Liu's color-segmentation equation, color image signalcan be transmitted with constant luminance and chromaticity, losslesshigh compression to conserve bandwidth at the same time;

(3) In color management system, D_(l)x_(l)y_(l) profile connection spaceis a universal path to achieve universal color mapping.

9. A method to rapidly calculate reference primary value c′m′y′ in RGBspace and Liu's 3-primary color clamping equation

Objective: existing method to convert scanned RGB data to CIEXYZ iscreating conversion polynomial with regression analysis. It incursobvious conversion error. Furthermore, regression analysis ismathematics approximation method and has nothing to do with gamutmapping which falls in field of visual psychology, so it does not accordwith gamut mapping rules. Based on colorimetry concept, this inventionconverts RGB into reference primary value c′m′y′ in RGB space, thenconverts reference primary c′m′y′ to scan primary value cmy in XYZ colorspace; finally plug cmy into scanning color prediction equation toconvert value RGB from scanner to value CIEXYZ accurately. In abovesteps, the method to convert RGB to twin primary value c′m′y′ is basedon Liu's 3-primary color clamping equation.

Format of Liu's 3-primary clamping equation:

Liu's 3-primary clamping equation inherits type property from Liu'scolor-segmentation equation, and has 3 sub-types: CMK, CYK, MYK. Thedecision to choose the correct equation type to convert color RGB ismade based on the minimum value of RGB. The 3 sub-types of Liu's3-primary clamping equation are listed as below:

$\left\{ {\begin{matrix}{{\lambda \; R} = \begin{matrix}{{\left( {1 - y_{dd}^{\prime}} \right)\left( {1 - m^{\prime}} \right)\left( {1 - c^{\prime}} \right)R_{w}} + {\left( {1 - y_{dd}^{\prime}} \right)\left( {1 - m^{\prime}} \right)c^{\prime}R_{c}} + {\left( {1 - y_{dd}^{\prime}} \right){m_{x}^{\prime}\left( {1 - c^{\prime}} \right)}R_{m}} + {{y_{dd}^{\prime}\left( {1 - m^{\prime}} \right)}\left( {1 - c^{\prime}} \right)R_{y}} +} \\{{y_{dd}^{\prime}{m^{\prime}\left( {1 - c^{\prime}} \right)}R_{r}} + {{y_{dd}^{\prime}\left( {1 - m^{\prime}} \right)}c^{\prime}R_{g}} + {\left( {1 - y_{dd}^{\prime}} \right)m^{\prime}c^{\prime}R_{b}} + {y_{dd}^{\prime}m^{\prime}c^{\prime}R_{s}}}\end{matrix}} \\{{\lambda \; G} = {{\left( {1 - y_{dd}^{\prime}} \right)\left( {1 - m^{\prime}} \right)\left( {1 - c^{\prime}} \right)G_{w}} + {\left( {1 - y_{dd}^{\prime}} \right)\left( {1 - m^{\prime}} \right)c^{\prime}G_{c}} + {\left( {1 - y_{dd}^{\prime}} \right){m^{\prime}\left( {1 - c^{\prime}} \right)}G_{m}} + \ldots + {y_{dd}^{\prime}m^{\prime}c^{\prime}G_{s}}}} \\{{\lambda \; B} = {{\left( {1 - y_{dd}^{\prime}} \right)\left( {1 - m^{\prime}} \right)\left( {1 - c^{\prime}} \right)B_{w}} + {\left( {1 - y_{dd}^{\prime}} \right)\left( {1 - m^{\prime}} \right)c^{\prime}B_{c}} + {\left( {1 - y_{dd}^{\prime}} \right){m^{\prime}\left( {1 - c^{\prime}} \right)}B_{m}} + \ldots + {y_{dd}^{\prime}m^{\prime}c^{\prime}B_{s}}}}\end{matrix}\left\{ {\begin{matrix}{{\lambda \; R} = {{\left( {1 - y^{\prime}} \right)\left( {1 - m_{dd}^{\prime}} \right)\left( {1 - c^{\prime}} \right)R_{w}} + {\left( {1 - y^{\prime}} \right)\left( {1 - m_{dd}^{\prime}} \right)c^{\prime}R_{c}} + {\left( {1 - y^{\prime}} \right){m_{dd}^{\prime}\left( {1 - c^{\prime}} \right)}R_{m}} + \ldots + {y^{\prime}m_{dd}^{\prime}c^{\prime}R_{s}}}} \\{{\lambda \; G} = {{\left( {1 - y^{\prime}} \right)\left( {1 - m_{dd}^{\prime}} \right)\left( {1 - c^{\prime}} \right)G_{w}} + {\left( {1 - y^{\prime}} \right)\left( {1 - m_{dd}^{\prime}} \right)c^{\prime}G_{c}} + {\left( {1 - y^{\prime}} \right){m_{dd}^{\prime}\left( {1 - c^{\prime}} \right)}G_{m}} + \ldots + {y^{\prime}m_{dd}^{\prime}c^{\prime}G_{s}}}} \\{{\lambda \; B} = {{\left( {1 - y^{\prime}} \right)\left( {1 - m_{dd}^{\prime}} \right)\left( {1 - c^{\prime}} \right)B_{w}} + {\left( {1 - y^{\prime}} \right)\left( {1 - m_{dd}^{\prime}} \right)c^{\prime}B_{c}} + {\left( {1 - y^{\prime}} \right){m_{dd}^{\prime}\left( {1 - c^{\prime}} \right)}B_{m}} + \ldots + {y^{\prime}m_{dd}^{\prime}c^{\prime}B_{s}}}}\end{matrix}\left\{ \begin{matrix}{{\lambda \; R} = {{\left( {1 - y^{\prime}} \right)\left( {1 - m^{\prime}} \right)\left( {1 - c_{dd}^{\prime}} \right)R_{w}} + {\left( {1 - y^{\prime}} \right)\left( {1 - m^{\prime}} \right)c_{dd}^{\prime}R_{c}} + {\left( {1 - y^{\prime}} \right){m_{x}^{\prime}\left( {1 - c_{dd}^{\prime}} \right)}R_{m}} + \ldots + {y^{\prime}m^{\prime}c_{dd}^{\prime}R_{s}}}} \\{{\lambda \; G} = {{\left( {1 - y^{\prime}} \right)\left( {1 - m^{\prime}} \right)\left( {1 - c_{dd}^{\prime}} \right)G_{w}} + {\left( {1 - y^{\prime}} \right)\left( {1 - m^{\prime}} \right)c_{dd}^{\prime}G_{c}} + {\left( {1 - y^{\prime}} \right){m^{\prime}\left( {1 - c_{dd}^{\prime}} \right)}G_{m}} + \ldots + {y^{\prime}m^{\prime}c_{dd}^{\prime}G_{s}}}} \\{{\lambda \; B} = {{\left( {1 - y^{\prime}} \right)\left( {1 - m^{\prime}} \right)\left( {1 - c_{dd}^{\prime}} \right)B_{w}} + {\left( {1 - y^{\prime}} \right)\left( {1 - m^{\prime}} \right)c_{dd}^{\prime}B_{c}} + {\left( {1 - y^{\prime}} \right){m^{\prime}\left( {1 - c_{dd}^{\prime}} \right)}B_{m}} + \ldots + {y^{\prime}m^{\prime}c_{dd}^{\prime}B_{s}}}}\end{matrix} \right.} \right.} \right.$

In above, λ is Liu's color appearance keeping parameter, c_(dd)′,m_(dd)′, y_(dd)′ are gray core parameters from scanner gray balancepower function expression. If type CMK of Liu's 3-primary color clampingequation is chosen to perform conversion for color RGB, only gray corevalue y_(dd) needs to be calculated, then dynamically plug y_(dd) intoLiu's 3-primary color clamping equation to get a quadratic equation withonly 3 unknown values λ, c′, m′, and can be solved quickly withanalytical algorithm. An equally important thing is: with joint effectof Liu's color appearance keeping parameter λ and gray core y_(dd), typeCMK of Liu's 3-primary color clamping equation grant primary value c′,m′, y′=y_(dd) with characteristic of reference primary value. If typeCYK or MYK of Liu's 3-primary color clamping equation is chosen, thenrepeat the same procedure as above except substitute gray core y_(dd)′with m_(dd) or c_(dd).

10. A method to convert scanned color from RGB color space to CIEXYZcolor space:

Objective: scanner captures RGB data reflected by color target image viaCCD sensor, we can assume scanner's CCD sensor has lineal response toamount of censored light; while XYZ value generated by scanned colortarget is non-lineal. The following method explains how to performRGB-CIEXYZ color space conversion for scanned RGB data in accordancewith colorimetry theory.

Procedure:

(1) Observe the RGB tristimulus value obtained by scanner and choose thesegmentation equation based on the minimum value in R, G, B. Theselection criteria are: if R is minimum value, choose GBK segmentationequation; if G is minimum value, choose RBK segmentation equation; if Bis minimum value, choose RGK segmentation equation;

(2) Plug tristimulus value RGB captured by scanner in left side ofcolor-segmentation equation (type GBK) to solve black value p;

(3) Plug black value p into gray scale format of Liu'scolor-segmentation equation to get gray luminance G_(l):G_(l)=G_(w)(1−p)+p G_(sk)

(4) Convert G_(l) into density value D_(rgb): D_(rgb)=lg(G_(wo)/G_(l));in which G_(wo) is luminance value measured on white area of scannedcolor target;

(5) Plug D_(rgb) into scanner's gray balance power function to get graycore value c_(dd)′;

(6) Plug gray core value c_(dd)′ into Liu's color-segmentation equation(type GBK) to get component primary value c′, m′, y′ of color RGB;

(7) Plug primary value c′, m′, y′ into scanner's conversion formulac′m′y′-cmy to calculate reference primary value c, m, y;

(8) Plug reference primary value c, m, y into scanner color-matchingequation to get target value XYZ.

11. A method to create D_(l)x_(l)y_(l) profile connection color spacefor digital camera or digital television camera

Objective: digital camera, scanner or other image input device can beseen as composed of CCD sensor device's RGB-CIEXYZ coordinatesconversion module and color target's nonlinear reflection processmodule. This concept helps to solve the problem accurately, easily andmeet the mapping requirement.

Procedure: although digital camera and digital television camera usedifferent CCD device, D_(l)x_(l)y_(l) profile connection color space canbe created using same format of Liu's color-segmentation equation; thesteps are as follows:

(1) After thoroughly considering general demands of current multimediadevices, a set of generally recognized chromaticity coordinates forcolor-matching three primary colors RGB is created, let chromaticitycoordinates of white dot equal to that of reference white D65. Below isthe matrix equation and luminance equation to perform the coordinateconversion from RGB to XYZ. The conversion result is accurate becausethis method is the geometric linear conversion of the same color indifferent color gauging system.

$\begin{bmatrix}X \\Y \\Z\end{bmatrix} = {\begin{bmatrix}a_{11} & a_{a\; 12} & a_{13} \\a_{21} & a_{a\; 22} & a_{23} \\a_{31} & a_{a\; 32} & a_{33}\end{bmatrix}\begin{bmatrix}R \\G \\B\end{bmatrix}}$ Y = a₂₁R + a₂₁G + a₂₁B

Plug a random RGB value captured by CCD image sensor component into theabove RGB-XYZ matrix equation to perform the RGB-XYZ color spacecoordinate data conversion.

(2) Below is function on creating gray scale format of Liu'ssegmentation equation for

X _(greyi) =X _(sk)·(1−Y)+Y·X _(sw)

standard monitor: Y_(greyi)=Y_(sk)·(1−Y)+Y·Y_(sw)

Z _(greyi) =Z _(sk)·(1−Y)+Y·Z _(sw)

In which, X_(gray)Y_(gray)Z_(gray) is tristimulus of pure gray ofstandard monitor,

-   -   X_(w)Y_(w)Z_(w) is tristimulus of white point of standard        monitor,    -   X_(k)Y_(k)Z_(k) is tristimulus of black point of standard        monitor;

(3) Convert gray scale luminance value Y_(gray) into gray density D_(l)as D_(l)=D_(n)=lg(Y_(w)/Y_(gray))

(4) Calculate chromaticity coordinates of color XYZ based on XYZ on leftside of RGB-XYZ matrix equation: with XYZ values, x_(l), y_(l) inD_(l)x_(l)y_(l) profile connection color space can be calculated asfollows:

x _(l) =x _(n) =X/(X+Y+Z), y _(l) =y _(n) =Y/(X+Y+Z)

After the above 4 steps, scan color XYZ is converted into profileconnection space value D_(n)X_(n)Y_(n).

12. A method to map parameter D_(l) of profile connection space to graytone parameter of target device gamut

Objective: create a universal method to seamlessly link parameter D_(l)of profile connection space with gray tone density of target colorgamut;

Procedure: following above steps, we have got gray tone density arrays[D_(ni)], [D_(oi)], [D_(pi)], [D_(qwi)], [D_(qki)] in device gamut (thedevice can be digital camera, digital video camera, scanner, monitor,printer or press). Based on mapping relationships between these arrays,we can get density mapping function with input device's gray tonedensity array as independent variable, output device's gray tone densityarray as dependent variable. Density mapping functions are as follows:

(1) Map digital camera's gray tone density array [D_(ni)] to gamut ofnormally-white-monitor (computer monitor): do power function fittingwith camera's gray tone density array [D_(ni)] as independent variableand normally-white-monitor's gray tone density array [D_(qwi)] asdependent variable to get gray density mapping function between digitalcamera and computer monitor:

D_(qw)=D_(n) ^(γ) ^(qw)

(2) Map television camera's gray tone density array [D_(ni)] to gamut ofnormally-black-monitor (television monitor): perform power functionfitting with television camera's gray tone density array [D_(ni)] asindependent variable and normally-black-monitor's gray tone densityarray [D_(qki)] as dependent variable to get gray tone density mappingfunction between television camera and television monitor: D_(qk)=D_(n)^(γ) ^(qk)

(3) Map scanner's gray tone density array [D_(oi)] to gamut of printeror press: perform power function fitting with scanner's gray tonedensity array [D_(oi)] as independent variable and printing device'sgray tone density array [D_(pi)] as dependent variable to get graydensity mapping function between scanner and printer (or press):D_(p)=D_(o) ^(γ) ^(po)

(4) Map printer or press's gray tone density array [D_(pi)] to gamut ofdisplay device: perform power function fitting with printer or press'sgray tone density array [D_(pi)] as independent variable andnormally-white-monitor's gray tone density array [D_(qwi)] as dependentvariable to get gray density mapping function between printer (or press)and monitor: D_(qw)=D_(p) ^(γ) ^(qwp)

13. A method to map color from source device to gamut of target device

Objective: Create a method to transform D_(l)x_(l)y_(l) in profileconnection space to target gamut without data loss.

(1) A method to map color obtained by scanner to gamut of printer device

Method: gamut mapping between scanner and printer device is performed bytransferring D_(o), x_(o), y_(o) of profile connection space to gamut ofprinter device with the help of Liu's scanner-printer gamut mappingequation.

Procedure:

Step 1, Based on scanned RGB, choose proper sub-type of Liu'sscanner-printer mapping equation: Liu's scanner-printer mapping equationhas 3 sub-types: CM K, CYK and MYK. There is one-to-one relationshipbetween them and sub-types of Liu's color-segmentation equation;following are the equation of 3 sub-types:

$\left\{ {\begin{matrix}{{\left( {x_{o}/y_{o}} \right)Y_{p}} = \begin{matrix}{{\left( {1 - c_{dd}} \right)\left( {1 - m} \right)\left( {1 - y} \right)\left( {1 - k_{dd}} \right)X_{w}} + {{c_{dd}\left( {1 - m} \right)}\left( {1 - y} \right)\left( {1 - k_{dd}} \right)X_{c}} + {\left( {1 - c_{dd}} \right){m\left( {1 - y} \right)}\left( {1 - k_{dd}} \right)X_{m}} +} \\{{\left( {1 - c_{dd}} \right)\left( {1 - m} \right){y\left( {1 - k_{dd}} \right)}X_{y}} + {\left( {1 - c_{dd}} \right){{my}\left( {1 - k_{dd}} \right)}X_{r}} + {{c_{dd}\left( {1 - m} \right)}{y\left( {1 - k_{dd}} \right)}X_{g}} + {c_{dd}{m\left( {1 - y} \right)}\left( {1 - k_{dd}} \right)X_{b}} +} \\{{c_{dd}{{my}\left( {1 - k_{dd}} \right)}X_{s}} + {\left( {1 - c_{dd}} \right)\left( {1 - m} \right)\left( {1 - y} \right)k_{dd}X_{k}} + {{c_{dd}\left( {1 - m} \right)}\left( {1 - y} \right)k_{dd}X_{ck}} + {\left( {1 - c_{dd}} \right){m\left( {1 - y} \right)}k_{dd}X_{mk}} +} \\{{\left( {1 - c_{dd}} \right)\left( {1 - m} \right)y\; k_{dd}X_{yk}} + {\left( {1 - c_{dd}} \right)m\; y\; k_{dd}X_{tk}} + {{c_{dd}\left( {1 - m} \right)}y\; k_{dd}X_{gk}} + {c_{dd}{m\left( {1 - y} \right)}k_{dd}X_{bk}} + {c_{dd}m\; y\; k_{dd}X_{sk}}}\end{matrix}} \\{Y_{p} = {{\left( {1 - c_{dd}} \right)\left( {1 - m} \right)\left( {1 - y} \right)\left( {1 - k_{dd}} \right)Y_{w}} + {{c_{dd}\left( {1 - m} \right)}\left( {1 - y} \right)\left( {1 - k_{dd}} \right)Y_{c}} + {\left( {1 - c_{dd}} \right){m\left( {1 - y} \right)}\left( {1 - k_{dd}} \right)Y_{m}} + \ldots + {c_{dd}m\; y\; k_{dd}Y_{ck}}}} \\{{\left\lbrack {\left( {1 - x_{o} - y_{o}} \right)/y_{o}} \right\rbrack Y_{p}} = {{\left( {1 - c_{dd}} \right)\left( {1 - m} \right)\left( {1 - y} \right)\left( {1 - k_{dd}} \right)Z_{w}} - {{c_{dd}\left( {1 - m} \right)}\left( {1 - y} \right)\left( {1 - k_{dd}} \right)Z_{c}} + \ldots + {c_{dd}m\; y\; k_{dd}Z_{sk}}}}\end{matrix}\left\{ {\begin{matrix}{{\left( {x_{o}/y_{o}} \right)Y_{p}} = {{\left( {1 - c} \right)\left( {1 - m_{dd}} \right)\left( {1 - y} \right)\left( {1 - k_{dd}} \right)X_{w}} + {{c\left( {1 - m_{dd}} \right)}\left( {1 - y} \right)\left( {1 - k_{dd}} \right)X_{c}} + \ldots + {c\; m_{dd}y\; k_{dd}X_{sk}}}} \\{Y_{p} = {{\left( {1 - c} \right)\left( {1 - m_{dd}} \right)\left( {1 - y} \right)\left( {1 - k_{dd}} \right)Y_{w}} + {{c\left( {1 - m_{dd}} \right)}\left( {1 - y} \right)\left( {1 - k_{dd}} \right)Y_{c}} + {\left( {1 - c} \right){m_{dd}\left( {1 - y} \right)}\left( {1 - k_{dd}} \right)Y_{m}} + \ldots + {c\; m_{dd}y\; k_{dd}Y_{sk}}}} \\{{\left\lbrack {\left( {1 - x_{o} - y_{o}} \right)/y_{o}} \right\rbrack Y_{p}} = {{\left( {1 - c} \right)\left( {1 - m_{dd}} \right)\left( {1 - y} \right)\left( {1 - k_{dd}} \right)Z_{w}} + {{c\left( {1 - m_{dd}} \right)}\left( {1 - y} \right)\left( {1 - k_{dd}} \right)Z_{c}} + \ldots + {c\; m_{dd}y\; k_{dd}Z_{sk}}}}\end{matrix}\left\{ \begin{matrix}{{\left( {x_{o}/y_{o}} \right)Y_{p}} = {{\left( {1 - c} \right)\left( {1 - m} \right)\left( {1 - y_{dd}} \right)\left( {1 - k_{dd}} \right)X_{w}} + {{c\left( {1 - m} \right)}\left( {1 - y_{dd}} \right)\left( {1 - k_{dd}} \right)X_{c}} + \ldots + {c\; m\; y_{dd}\; k_{dd}X_{sk}}}} \\{Y_{p} = {{\left( {1 - c} \right)\left( {1 - m} \right)\left( {1 - y_{dd}} \right)\left( {1 - k_{dd}} \right)Y_{w}} + {{c\left( {1 - m} \right)}\left( {1 - y_{dd}} \right)\left( {1 - k_{dd}} \right)Y_{c}} + {\left( {1 - c} \right){m\left( {1 - y_{dd}} \right)}\left( {1 - k_{dd}} \right)Y_{m}} + \ldots + {c\; m\; y_{dd}k_{dd}Y_{sk}}}} \\{{\left\lbrack {\left( {1 - x_{o} - y_{o}} \right)/y_{o}} \right\rbrack Y_{p}} = {{\left( {1 - c} \right)\left( {1 - m} \right)\left( {1 - y_{dd}} \right)\left( {1 - k_{dd}} \right)Z_{w}} + {{c\left( {1 - m} \right)}\left( {1 - y_{dd}} \right)\left( {1 - k_{dd}} \right)Z_{c}} + \ldots + {c\; m\; y_{dd}\; k_{dd}Z_{sk}}}}\end{matrix} \right.} \right.} \right.$

For the sake of simplicity, we only take MYK equation as an example.Plug scanner's chromaticity coordinates x_(o) and y_(o) fromD_(l)x_(l)y_(l) profile connection space to left of the equation,parameter Y_(p) is an unknown luminance parameter. Y_(p) has theclamping luminance characteristic and is called Liu's clamping luminanceThe left side of the equation shows scanned output color D_(o)x_(o)y_(o)is going to be printed as a new color after processed by scan-printmapping equation. Let's say X_(p), Y_(p), Z_(p) is the new color;X_(p)Y_(p)Z_(p) actually is the value from left side of the equation.CMYK is the driving value for displaying color X_(p), Y_(p), Z_(p).

X _(p)=(x _(o) /y _(o))Y _(p) , Y _(p) =Y _(p) , Z _(p)=[(1−x _(o) −y_(o))/y _(o) ]Y _(p)

Step 2, Map scanner color space's gray tone density D_(o) to gray tonedensity D_(p) of printer device's gamut:

a, let D_(p)=D_(o) ^(γ) ^(po)

b, Calculate Gamma correction density: let D_(p)′=D_(p) ^(1/γ) ^(p) ²=(D_(o) ^(γ) ^(po) )^(1/γ) ^(p) ² =D_(o) ^(γ) ^(po) ^(/γ) ^(p) ²

c, Substitute D_(p) in printer's gray balance polynomial with Gammacorrection density D_(p)′ to get Liu's 4-primary color printinganti-Gamma gray balance polynomial, with which we can calculate Gammacorrected gray balance component value c_(p), m_(p), y_(p) and graysubstitute parameter k_(p); c_(p), m_(p), y_(p), k_(p) can be used asgray cores;

c _(p) =c _(dd) =a ₀ +a ₁ D _(p) ′+a ₂ D _(p)′² +a ₃ D _(p)′³ + . . . ,m _(p) =m _(dd) =b ₀ +b ₁ D _(p) ′+b ₂ D _(p)′² +b ₃ D _(p)′³+ . . .

y _(p) =y _(dd) =c ₀ +c ₁ D _(p) ′+c ₂ D _(p)′² +c ₃ D _(p)′³ + . . . ,k _(p) =k _(dd) =d ₀ +d ₁ D _(p) ′+d ₂ D _(p)′² +d ₃ D _(p)′³+ . . .

If CMK or CYK type equation is chosen, k_(dd), m_(dd) or k_(dd), y_(dd)need to be calculated instead of k_(dd), c_(dd);

d, based on chosen sub-type of Liu's scanner printer mapping equation,plug k_(dd), c_(dd) (or k_(dd), m_(dd), or k_(dd), y_(dd)) into rightside of chosen sub-type of 4-primary color mapping equation;

As shown above, the two new symbols c_(dd), k_(dd) on the right side ofthe equation are not variables. They are values provided by cmyk graybalance polynomial. For type MYK equation, c_(dd)=c, which comes fromprinter gray balance equation. The independent variable in gray balancepolynomial turns into D_(p). The gray color parameter D_(o) fromD_(o)x_(o)y_(o) profile connection space is related to Liu's scannerprinter equation mapping equation via D_(p) and c_(dd). k_(dd) is graycomponent substitute value of the color to be transformed. The variableof the equation is m, y and Y_(p), which helps to produce the colorX_(p)Y_(p)Z_(p) on the left of the equation. Similar to Liu's primaryclamping equation, variable Y_(p) also acts as color appearance keepingcoefficient. It ensures the new color X_(p)Y_(p)Z_(p) and color XYZ fromscanner output has the same color appearance with consistent visualeffect. The three primary value c=c_(dd), m, y and c_(dd) calculatedusing type MYK of Liu's scanner-printer mapping equation are Gammacorrected reference primary value.

Step 3, Calculate driving value CMYK which makes up color X_(p), Y_(p),Z_(p) based on 3 primary value c=c_(dd), m, y derived from Liu'sscanner-printer mapping equation. In the equation, reference primarycmyk is the function of driving input value d_(c) d_(m) d_(y) d_(k).Printer reference primary cmyk can be converted into driving input valueCMYK as follows:

C=d _(cp) =D _(p) ^(1/γ) ^(p) =(c _(p) ^(1/γ) ^(cd) )^(1/γ) ^(p) =c^(1/(γ) ^(cd) ^(γ) ^(p)) , M=d _(mp) =D _(p) ^(1/γ) ^(p) =(m _(p) ^(1/γ)^(md) )^(1/γ) ^(p) =m ^(1/(γ) ^(md) ^(γ) ^(p))

Y=d _(yp) =D _(p) ^(1/γ) ^(p) =(y _(p) ^(1/γ) ^(yd) )^(1/γ) ^(p) =y^(1/(γ) ^(yd) ^(γ) ^(p)) , K=d _(kp) =D _(p) ^(1/γ) ^(p) =(k _(p) ^(1/γ)^(kd) )^(1/γ) ^(p) =k _(p) ^(1/(γ) ^(kd) ^(γ) ^(p))

(2) A method to map color XYZ captured by digital camera to color gamutof normally-white-monitor

Method: gamut mapping between digital camera and display device isperformed by transferring profile connection space parameter D_(n),x_(n), y_(n) to display device with the help of digital camera-monitormapping equation.

Procedure:

Step 1, map digital camera color space's gray tone density D_(n) to graytone density D_(q) of display device gamut:

a, Let D_(qw)=D_(n) ^(γ) ^(qnw) b, Calculate Gamma correction densityD_(qw)′=D_(qw) ^(1/γ) ^(qw) ²

c, Plug Gamma correction density D_(qw)′ into normally-white-monitor'sgray balance component primary power function to calculate gray balancecomponent primary value (used as gray core) and gray substituteparameter k_(p):

r _(dd)=(D _(qw)′)^(γ) ^(rd) =(D _(qw) ^(1/γ) ^(qw) ² )^(γ) ^(rd) =D_(qw) ^(γ) ^(rd) ^(/γ) ^(qw) ² , g _(dd)=(D _(qw)′)^(γ) ^(gd) =D _(qw)^(γ) ^(gd) ^(/γ) ^(qw) ² , b _(dd)=(D _(qw)′)^(γ) ^(bd) =D _(qw) ^(γ)^(bd) ^(/γ) ^(qw) ²

Step 2, map chromaticity coordinates x_(n) and y_(n) of input deviceprofile connection space to normally-white-monitor, then finish mappingusing Liu's digital camera-monitor mapping equation:

Liu's digital camera-monitor mapping equation has 3 sub-types: r_(dd)gb,rg_(dd)b, rgb_(dd). There is one-to-one relationship between them andsub-types of Liu's color-segmentation equation; following are the 3sub-types:

$\left\{ {\begin{matrix}{{\left( {x_{n}/y_{n}} \right)Y_{qw}} = \begin{matrix}{{\left( {1 - r_{dd}} \right)\left( {1 - g} \right)\left( {1 - b} \right)X_{w}} + {\left( {1 - r_{dd}} \right){g_{d}\left( {1 - b} \right)}X_{m}} + {\left( {1 - r_{dd}} \right)\left( {1 - g} \right)b\; X_{y}} +} \\{{\left( {1 - r_{dd}} \right)g\; b\; X_{r}} + {{r_{dd}\left( {1 - g} \right)}\left( {1 - b} \right)X_{c}} + {{r_{dd}\left( {1 - g} \right)}b\; X_{g}} + {r_{dd}{g\left( {1 - b} \right)}X_{b}} + {r_{dd}g\; b\; X_{sk}}}\end{matrix}} \\{Y_{qw} = {{\left( {1 - r_{dd}} \right)\left( {1 - g} \right)\left( {1 - b} \right)Y_{w}} + {\left( {1 - r_{dd}} \right){g\left( {1 - b} \right)}Y_{m}} + {\left( {1 - r_{dd}} \right)\left( {1 - g} \right)b\; Y_{y}} + \ldots + {r_{dd}g\; b\; Y_{sk}}}} \\{{\left\lbrack {\left( {1 - x_{n} - y_{n}} \right)/y_{n}} \right\rbrack Y_{qw}} = {{\left( {1 - r_{dd}} \right)\left( {1 - g} \right)\left( {1 - b} \right)Z_{w}} + {\left( {1 - r_{dd}} \right){g\left( {1 - b} \right)}Z_{m}} + {\left( {1 - r_{dd}} \right)\left( {1 - g} \right)b\; Z_{y}} + \ldots + {r_{dd}g\; b\; Z_{sk}}}}\end{matrix}\left\{ {\begin{matrix}{{\left( {x_{n}/y_{n}} \right)Y_{qw}} = \begin{matrix}{{\left( {1 - r} \right)\left( {1 - g_{dd}} \right)\left( {1 - b} \right)X_{w}} + {\left( {1 - r} \right){g_{dd}\left( {1 - b} \right)}X_{m}} + {\left( {1 - r} \right)\left( {1 - g_{dd}} \right)b\; X_{y}} +} \\{{\left( {1 - r} \right)g_{dd}\; b\; X_{r}} + {{r\left( {1 - g_{dd}} \right)}\left( {1 - b} \right)X_{c}} + {{r\left( {1 - g_{dd}} \right)}b\; X_{g}} + {r_{d}\; {g\left( {1 - b} \right)}X_{b}} + {r\; g_{dd}\; b\; X_{sk}}}\end{matrix}} \\{Y_{qw} = {{\left( {1 - r} \right)\left( {1 - g_{dd}} \right)\left( {1 - b} \right)Y_{w}} + {\left( {1 - r} \right){g_{dd}\left( {1 - b} \right)}Y_{m}} + {\left( {1 - r} \right)\left( {1 - g_{dd}} \right)b\; Y_{y}} + \ldots + {r\; g_{dd}\; b\; Y_{sk}}}} \\{{\left\lbrack {\left( {1 - x_{n} - y_{n}} \right)/y_{n}} \right\rbrack Y_{qw}} = {{\left( {1 - r} \right)\left( {1 - g_{dd}} \right)\left( {1 - b} \right)Z_{w}} + {\left( {1 - r} \right){g_{dd}\left( {1 - b} \right)}Z_{m}} + {\left( {1 - r} \right)\left( {1 - g_{dd}} \right)b\; Z_{y}} + \ldots + {r\; g_{dd}\; b\; Z_{sk}}}}\end{matrix}\left\{ \begin{matrix}{{\left( {x_{n}/y_{n}} \right)Y_{qw}} = \begin{matrix}{{\left( {1 - r} \right)\left( {1 - g} \right)\left( {1 - b_{dd}} \right)X_{w}} + {\left( {1 - r} \right){g\left( {1 - b_{dd}} \right)}X_{m}} + {\left( {1 - r} \right)\left( {1 - g} \right)b_{dd}\; X_{y}} +} \\{{\left( {1 - r} \right)g\; b_{dd}X_{r}} + {{r\left( {1 - g} \right)}\left( {1 - b_{dd}} \right)X_{c}} + {{r\left( {1 - g} \right)}b_{dd}\; X_{g}} + {r\; {g\left( {1 - b_{dd}} \right)}X_{b}} + {r\; g\; b_{dd}X_{sk}}}\end{matrix}} \\{Y_{qw} = {{\left( {1 - r} \right)\left( {1 - g} \right)\left( {1 - b_{dd}} \right)Y_{w}} + {\left( {1 - r} \right){g\left( {1 - b_{dd}} \right)}Y_{m}} + {\left( {1 - r} \right)\left( {1 - g} \right)b_{dd}\; Y_{y}} + \ldots + {r\; g\; b_{dd}\; Y_{sk}}}} \\{{\left\lbrack {\left( {1 - x_{n} - y_{n}} \right)/y_{n}} \right\rbrack Y_{qw}} = {{\left( {1 - r} \right)\left( {1 - g} \right)\left( {1 - b_{dd}} \right)Z_{w}} + {\left( {1 - r} \right){g\left( {1 - b_{dd}} \right)}Z_{m}} + {\left( {1 - r} \right)\left( {1 - g} \right)b_{dd}\; Z_{y}} + \ldots + {r\; g\; b_{dd}\; Z_{sk}}}}\end{matrix} \right.} \right.} \right.$

Choose a sub-type from r_(dd)gb, rg_(dd)b, rgb_(d), plug r_(dd), g_(dd),b_(dd) (calculated in step 1) into right side of equation; plugx_(n)y_(n) and Liu's clamping luminance Y_(qw) into left side ofequation to solve Gamma corrected and gamut mapped reference primaryvalue r,g,b and luminance value Y_(qw).

In above, driving input value d_(r)d_(g)d_(b) is also function ofreference primary value rgb. Hence we get solution to convertnormally-white-monitor reference primary value rgb to driving inputvalue RGB:

R=d _(qw) =D _(qw) ^(1/γ) ^(qw) =(r _(qw) ^(1/γ) ^(rdw) )^(1/γ) ^(qw) =r^(1/(γ) ^(rd) ^(γ) ^(qw)) G=d _(qw) =D _(qw) ^(1/γ) ^(qw) =(g _(qw)^(1/γ) ^(gdw) )^(1/γ) ^(qw) =g ^(1/(γ) ^(gd) ^(γ) ^(qw)) B=d _(qw) =D_(qw) ^(1/γ) ^(qw) =(b _(qw) ^(1/γ) ^(bdw) )^(1/γ) ^(qw) =b ^(1/(γ)^(bd) ^(γ) ^(qw))

The 3 primary values calculated respectively using 3 different types ofLiu's comprehensive mapping equation are all Gamma corrected primaryvalue.

Let X_(qw)=(x_(n)/y_(n))Y_(t), Y_(qw)=Y_(t),Z_(qw)=[(1−x_(n)−y_(n))/y_(n)]Y_(t), the final display color is X_(qw),Y_(qw), Z_(qw) and RGB is driving value used to display color X_(qw),Y_(qw), Z_(qw).

(3) A method to map color XYZ captured by television video camera togamut of normally-black television monitor

Method: gamut mapping between television video camera and televisiondisplay device is performed by transferring profile connection spaceparameter D_(n), x_(n), y_(n) to display device with the help oftelevision video camera-television monitor mapping equation.

Procedure:

Step 1, Map gray tone density D_(n) of television video camera colorspace to gray tone density D_(qk) of display device gamut.

a, Let D_(qk)=D_(n) ^(γ) ^(qnk) b, Calculate Gamma correction density:let D_(qk)′=D_(qk) ^(1/γ) ^(qk) ²

c, Plug Gamma correction density D_(qk)′ into normally-black-monitor'sgray balance component primary power function to calculate gray balancecomponent primary value (used as gray core) and gray substituteparameter k_(dd):

r _(dd)=(D _(qk)′)^(γ) ^(rd) =(D _(qk) ^(1/γ) ^(qk) ² )^(γ) ^(rd) =D_(qk) ^(γ) ^(rd) ^(/γ) _(qk) ² , g _(dd)=(D _(qk)′)^(γ) ^(dg) =D _(qk)^(γ) ^(gd) ^(/γ) ^(qk) ² , b _(dd)=(D _(qk)′)^(γ) ^(bd) =D _(qk) ^(γ)^(bd) ^(/γ) ^(qk) ²

Step 2, Map chromaticity coordinates x_(n) and y_(n) of input deviceprofile connection space to television monitor, and then finish mappingwith Liu's mapping equation:

Liu's 4-primary color gamut mapping equation has 3 sub-types: r_(dd)gb,rg_(dd)b, rgb_(dd). There is one-to-one relationship between them andsub-types of Liu's color-segmentation equation; following are the 3sub-types:

$\left\{ {\begin{matrix}{{\left( {x_{n}/y_{n}} \right)Y_{qk}} = \begin{matrix}{{\left( {1 - r_{dd}} \right)\left( {1 - g} \right)\left( {1 - b} \right)X_{k}} + {\left( {1 - r_{dd}} \right){g_{d}\left( {1 - b} \right)}X_{m}} + {\left( {1 - r_{dd}} \right)\left( {1 - g} \right)b\; X_{y}} +} \\{{\left( {1 - r_{dd}} \right)g\; b\; X_{r}} + {{r_{dd}\left( {1 - g} \right)}\left( {1 - b} \right)X_{c}} + {{r_{dd}\left( {1 - g} \right)}b\; X_{g}} + {r_{dd}{g\left( {1 - b} \right)}X_{b}} + {r_{dd}g\; b\; X_{sw}}}\end{matrix}} \\{Y_{qk} = {{\left( {1 - r_{dd}} \right)\left( {1 - g} \right)\left( {1 - b} \right)Y_{k}} + {\left( {1 - r_{dd}} \right){g\left( {1 - b} \right)}Y_{m}} + {\left( {1 - r_{dd}} \right)\left( {1 - g} \right)b\; Y_{y}} + \ldots + {r_{dd}g\; b\; Y_{sw}}}} \\{{\left\lbrack {\left( {1 - x_{n} - y_{n}} \right)/y_{n}} \right\rbrack Y_{qk}} = {{\left( {1 - r_{dd}} \right)\left( {1 - g} \right)\left( {1 - b} \right)Z_{k}} + {\left( {1 - r_{dd}} \right){g\left( {1 - b} \right)}Z_{m}} + {\left( {1 - r_{dd}} \right)\left( {1 - g} \right)b\; Z_{y}} + \ldots + {r_{dd}g\; b\; Z_{sw}}}}\end{matrix}\left\{ {\begin{matrix}{{\left( {x_{n}/y_{n}} \right)Y_{qk}} = \begin{matrix}{{\left( {1 - r} \right)\left( {1 - g_{dd}} \right)\left( {1 - b} \right)X_{k}} + {\left( {1 - r} \right){g_{dd}\left( {1 - b} \right)}X_{m}} + {\left( {1 - r} \right)\left( {1 - g_{dd}} \right)b\; X_{y}} +} \\{{\left( {1 - r} \right)g_{dd}\; b\; X_{r}} + {{r\left( {1 - g_{dd}} \right)}\left( {1 - b} \right)X_{c}} + {{r\left( {1 - g_{dd}} \right)}b\; X_{g}} + {r_{d}\; {g\left( {1 - b} \right)}X_{b}} + {r\; g_{dd}\; b\; X_{sw}}}\end{matrix}} \\{Y_{qk} = {{\left( {1 - r} \right)\left( {1 - g_{dd}} \right)\left( {1 - b} \right)Y_{k}} + {\left( {1 - r} \right){g_{dd}\left( {1 - b} \right)}Y_{m}} + {\left( {1 - r} \right)\left( {1 - g_{dd}} \right)b\; Y_{y}} + \ldots + {r\; g_{dd}\; b\; Y_{sw}}}} \\{{\left\lbrack {\left( {1 - x_{n} - y_{n}} \right)/y_{n}} \right\rbrack Y_{qk}} = {{\left( {1 - r} \right)\left( {1 - g_{dd}} \right)\left( {1 - b} \right)Z_{k}} + {\left( {1 - r} \right){g_{dd}\left( {1 - b} \right)}Z_{m}} + {\left( {1 - r} \right)\left( {1 - g_{dd}} \right)b\; Z_{y}} + \ldots + {r\; g_{dd}\; b\; Z_{sw}}}}\end{matrix}\left\{ \begin{matrix}{{\left( {x_{n}/y_{n}} \right)Y_{qk}} = \begin{matrix}{{\left( {1 - r} \right)\left( {1 - g} \right)\left( {1 - b_{dd}} \right)X_{k}} + {\left( {1 - r} \right){g\left( {1 - b_{dd}} \right)}X_{m}} + {\left( {1 - r} \right)\left( {1 - g} \right)b_{dd}\; X_{y}} +} \\{{\left( {1 - r} \right)g\; b_{dd}X_{r}} + {{r\left( {1 - g} \right)}\left( {1 - b_{dd}} \right)X_{c}} + {{r\left( {1 - g} \right)}b_{dd}\; X_{g}} + {r\; {g\left( {1 - b_{dd}} \right)}X_{b}} + {r\; g\; b_{dd}X_{sw}}}\end{matrix}} \\{Y_{qk} = {{\left( {1 - r} \right)\left( {1 - g} \right)\left( {1 - b_{dd}} \right)Y_{k}} + {\left( {1 - r} \right){g\left( {1 - b_{dd}} \right)}Y_{m}} + {\left( {1 - r} \right)\left( {1 - g} \right)b_{dd}\; Y_{y}} + \ldots + {r\; g\; b_{dd}\; Y_{sw}}}} \\{{\left\lbrack {\left( {1 - x_{n} - y_{n}} \right)/y_{n}} \right\rbrack Y_{qk}} = {{\left( {1 - r} \right)\left( {1 - g} \right)\left( {1 - b_{dd}} \right)Z_{k}} + {\left( {1 - r} \right){g\left( {1 - b_{dd}} \right)}Z_{m}} + {\left( {1 - r} \right)\left( {1 - g} \right)b_{dd}\; Z_{y}} + \ldots + {r\; g\; b_{dd}\; Z_{sw}}}}\end{matrix} \right.} \right.} \right.$

Choose a sub-type from r_(dd)gb, rg_(dd)b, rgb_(dd); plug r_(dd),g_(dd), b_(dd) (calculated in step 1) into right side of equation; plugx_(n), y_(n) and Liu's clamping luminance Y_(qk) into left side ofequation to get Gamma corrected and gamut mapped reference primary valuergb and luminance value Y_(qk).

In above equations, driving input value d_(r)d_(g)d_(b) is also functionof reference primary value rgb.

Following is the method to convert television monitor reference primaryvalue rgb to driving input value RGB:)

R=d _(qk) =D _(qk) ^(1/γ) ^(qk) =(r _(qk) ^(1/γ) ^(rdk) )^(1/γ) ^(qk) =r^(1/(γ) ^(rd) ^(γ) ^(qk)) G=d _(qk) =D _(qk) ^(1/γ) ^(qk) =(g _(qk)^(1/γ) ^(gdk) )^(1/γ) ^(qk) =g ^(1/(γ) ^(gd) ^(γ) ^(qk)) B=d _(qk) =D_(qk) ^(1/γ) ^(qk) =(b _(qk) ^(1/γ) ^(bdk) )^(1/γ) ^(qk) =b ^(1/(γ)^(bd) ^(γ) ^(qk))

The 3 primary values calculated respectively with 3 sub-types of Liu'scomprehensive mapping equation are Gamma corrected primary value.

Let X_(qk)=(x_(n)/y_(n))Y_(n), Y_(qk)=Y_(n),Z_(qk)=[(1−x_(n)−y_(n))/y_(n)]Y_(n), the final display color is X_(qk),Y_(qk), Z_(qk). RGB is driving value used for displaying color X_(qk),Y_(qk), Z_(qk).

(4) Method to map color XYZ captured by printer device to gamut ofnormally-white-monitor

Method: gamut mapping between printer device and display device isperformed by mapping density parameter D_(p) of profile connection spaceto gray tone density parameter D_(pw) of target gamut, and transferringchromaticity coordinates x_(p) and y_(p) with the help ofprinter-monitor mapping equation.

Procedure:

Step 1, Map printer device color space's gray tone density D_(p) todisplay device gamut's gray tone density D_(q):

a. Let D_(qw)=D_(p) ^(γ) ^(qwp) ;

b. Calculate Gamma correction density: let D_(qw)′=d_(qw) ^(1/γ) ^(qw)=D_(qw) ^(1/γ) ^(qw) ² ;

Plug Gamma correction density D_(o)′ into printer's gray balancecomponent primary power function to calculate gray balance componentprimary value(used as gray core) and gray substitute parameter k_(p):

r _(dd)=(D _(qw)′)^(γ) ^(rd) =(D _(qw) ^(1/γ) ^(qk) ² )^(γ) ^(rd) =D_(qw) ^(γ) ^(rd) ^(/γ) ^(qk) ² , g _(dd)=(D _(qw)′)^(γ) ^(gd) =D _(qw)^(γ) ^(gd) ^(/γ) ^(qk) ² , b _(dd)=(D _(qw)′)^(γ) ^(bd) =D _(qw) ^(γ)^(bd) ^(/γ) ^(qk) ²

Only r_(dd) or g_(dd) or b_(dd) need to be calculated based on thechosen type of printer—normally-white-monitor gamut mapping equation.

c. Plug r_(dd), g_(dd) or b_(dd) into right side of chosen type ofprinter—normally-black-monitor gamut mapping equation;

Step 2, Map printing device profile connection space chromaticitycoordinates x_(p) and y_(p) to gamut of display device using Liu'sprinter—normally-white-monitor gamut mapping equation. The 3 sub-typesof equation r_(dd)gb, rg_(dd)b, rgb_(dd) are as follows:

$\left\{ {\begin{matrix}{{\left( {x_{p}/y_{p}} \right)Y_{qw}} = \begin{matrix}{{\left( {1 - r_{dd}} \right)\left( {1 - g} \right)\left( {1 - b} \right)X_{w}} + {\left( {1 - r_{dd}} \right){g_{d}\left( {1 - b} \right)}X_{m}} + {\left( {1 - r_{dd}} \right)\left( {1 - g} \right)b\; X_{y}} +} \\{{\left( {1 - r_{dd}} \right)g\; b\; X_{r}} + {{r_{dd}\left( {1 - g} \right)}\left( {1 - b} \right)X_{c}} + {{r_{dd}\left( {1 - g} \right)}b\; X_{g}} + {r_{dd}{g\left( {1 - b} \right)}X_{b}} + {r_{dd}g\; b\; X_{sk}}}\end{matrix}} \\{Y_{qw} = {{\left( {1 - r_{dd}} \right)\left( {1 - g} \right)\left( {1 - b} \right)Y_{w}} + {\left( {1 - r_{dd}} \right){g\left( {1 - b} \right)}Y_{m}} + {\left( {1 - r_{dd}} \right)\left( {1 - g} \right)b\; Y_{y}} + \ldots + {r_{dd}g\; b\; Y_{sk}}}} \\{{\left\lbrack {\left( {1 - x_{p} - y_{p}} \right)/y_{p}} \right\rbrack Y_{qw}} = {{\left( {1 - r_{dd}} \right)\left( {1 - g} \right)\left( {1 - b} \right)Z_{w}} + {\left( {1 - r_{dd}} \right){g\left( {1 - b} \right)}Z_{m}} + {\left( {1 - r_{dd}} \right)\left( {1 - g} \right)b\; Z_{y}} + \ldots + {r_{dd}g\; b\; Z_{sk}}}}\end{matrix}\left\{ {\begin{matrix}{{\left( {x_{p}/y_{p}} \right)Y_{qw}} = \begin{matrix}{{\left( {1 - r} \right)\left( {1 - g_{dd}} \right)\left( {1 - b} \right)X_{w}} + {\left( {1 - r} \right){g_{dd}\left( {1 - b} \right)}X_{m}} + {\left( {1 - r} \right)\left( {1 - g_{dd}} \right)b\; X_{y}} +} \\{{\left( {1 - r} \right)g_{dd}\; b\; X_{r}} + {{r\left( {1 - g_{dd}} \right)}\left( {1 - b} \right)X_{c}} + {{r\left( {1 - g_{dd}} \right)}b\; X_{g}} + {r_{d}\; {g\left( {1 - b} \right)}X_{b}} + {r\; g_{dd}\; b\; X_{sk}}}\end{matrix}} \\{Y_{qw} = {{\left( {1 - r} \right)\left( {1 - g_{dd}} \right)\left( {1 - b} \right)Y_{w}} + {\left( {1 - r} \right){g_{dd}\left( {1 - b} \right)}Y_{m}} + {\left( {1 - r} \right)\left( {1 - g_{dd}} \right)b\; Y_{y}} + \ldots + {r\; g_{dd}\; b\; Y_{sk}}}} \\{{\left\lbrack {\left( {1 - x_{p} - y_{p}} \right)/y_{p}} \right\rbrack Y_{qw}} = {{\left( {1 - r} \right)\left( {1 - g_{dd}} \right)\left( {1 - b} \right)Z_{w}} + {\left( {1 - r} \right){g_{dd}\left( {1 - b} \right)}Z_{m}} + {\left( {1 - r} \right)\left( {1 - g_{dd}} \right)b\; Z_{y}} + \ldots + {r\; g_{dd}\; b\; Z_{sk}}}}\end{matrix}\left\{ \begin{matrix}{{\left( {x_{p}/y_{p}} \right)Y_{qw}} = \begin{matrix}{{\left( {1 - r} \right)\left( {1 - g} \right)\left( {1 - b_{dd}} \right)X_{w}} + {\left( {1 - r} \right){g\left( {1 - b_{dd}} \right)}X_{m}} + {\left( {1 - r} \right)\left( {1 - g} \right)b_{dd}\; X_{y}} +} \\{{\left( {1 - r} \right)g\; b_{dd}X_{r}} + {{r\left( {1 - g} \right)}\left( {1 - b_{dd}} \right)X_{c}} + {{r\left( {1 - g} \right)}b_{dd}\; X_{g}} + {r\; {g\left( {1 - b_{dd}} \right)}X_{b}} + {r\; g\; b_{dd}X_{sk}}}\end{matrix}} \\{Y_{qw} = {{\left( {1 - r} \right)\left( {1 - g} \right)\left( {1 - b_{dd}} \right)Y_{w}} + {\left( {1 - r} \right){g\left( {1 - b_{dd}} \right)}Y_{m}} + {\left( {1 - r} \right)\left( {1 - g} \right)b_{dd}\; Y_{y}} + \ldots + {r\; g\; b_{dd}\; Y_{sk}}}} \\{{\left\lbrack {\left( {1 - x_{p} - y_{p}} \right)/y_{p}} \right\rbrack Y_{qw}} = {{\left( {1 - r} \right)\left( {1 - g} \right)\left( {1 - b_{dd}} \right)Z_{w}} + {\left( {1 - r} \right){g\left( {1 - b_{dd}} \right)}Z_{m}} + {\left( {1 - r} \right)\left( {1 - g} \right)b_{dd}\; Z_{y}} + \ldots + {r\; g\; b_{dd}\; Z_{sk}}}}\end{matrix} \right.} \right.} \right.$

In above equation, driving input value d_(r)d_(g)d_(b) is also functionof reference primary value rgb;

hence normally-white-monitor reference primary value rgb can beconverted to driving input value RGB as follows:

R=d _(qw) =D _(qw) ^(1/γ) ^(qw) =(r _(qw) ^(1/γ) ^(rdw) )^(1/γ) ^(qw) =r^(1/(γ) ^(rd) ^(γ) ^(qw)) G=d _(qw) =D _(qw) ^(1/γ) ^(qw) =(g _(qw)^(1/γ) ^(gdw) )^(1/γ) ^(qw) =g ^(1/(γ) ^(gd) ^(γ) ^(qw)) B=d _(qw) =D_(qw) ^(1/γ) ^(qw) =(b _(qw) ^(1/γ) ^(bdw) )^(1/γ) ^(qw) =b ^(1/(γ)^(bd) ^(γ) ^(qw))

3 primary values calculated respectively using 3 sub-types of Liu'scomprehensive mapping equation are all Gamma corrected primary values.

Let X_(qw)=(x_(n)/y_(n))Y_(t), Y_(qw)=Y_(t),Z_(qw)=[(1−x_(n)−y_(n))/y_(n)]Y_(t), and final display color is X_(qw),Y_(qw), Z_(qw) and RGB is driving value required for displaying colorX_(qw), Y_(qw), Z_(qw).

IV DESCRIPTION OF DRAWING

FIG. 1, Flow diagram of gamut mapping from scanner to printer devicecolor space, and from printer device color space to monitor device colorspace:

Predicted color based on color target scanned by scanner→transformpredicted color to Liu's D_(l)x_(l)y_(l) profile connectionspace→transform to printer device color space→transform to displaydevice color space;

FIG. 2, Flow diagram of converting color RGB captured by digital camerainto XYZ and calculating computer monitor's driving value RGB based onXYZ;

FIG. 3, Flow diagram of converting color RGB captured by digital videocamera or television camera into XYZ value and calculating televisionmonitor driving value RGB based on XYZ.

V. APPLICATION STEPS

1. Refer to FIG. 1, the following mapping flow is used to highlight thisinvention's generality, creativity and efficiency features: “predictedcolor XYZ based on scanned RBG from scanner→transform color XYZ to Liu'sD_(l)x_(l)y_(l) profile connection space→segment color XYZ to getdriving input value CMYK→produce pre-proofing color image onnormally-white-monitor”

(1) Convert R,G,B into tristimulus values X,Y,Z using the methodmentioned in this invention;

(2) Observe tristimulus value R,G,B captured by scanner's CCD device,choose proper RGB segmentation equation based on the smallest value inR,G,B.

The selection criteria are:

if B has smallest value, use Liu's scan segmentation equation-type MYK;

if R has smallest value, use Liu's scan segmentation equation-type CYK;

if G has smallest value, use Liu's scan segmentation equation-type CMK;

We will take type MYK of Liu's scan segmentation equation as example,other two scenarios can be done in similar manner;

(3) Plug target value XYZ on left side of Liu's scan segmentationequation (type MYK), solve the equation to get black vale p;

(4) Plug black value p into gray scale format of Liu's scan segmentationequation to get gray luminance tristimulus value Y_(l) asY_(l)=Y_(wo)(1−p)+p Y_(sk);

(5) Convert Y_(l) to density value D_(o): letD_(o)=D_(l)=lg(Y_(wo)/Y_(l)), in which Y_(wo) is luminance tristimulusvalue measured on white area of scan target;

(6) Convert tristimulus value XYZ into chromaticity coordinates x_(o)and y_(o), as follows:

$\begin{matrix}{x_{o} = {x_{l} = \frac{X}{X + Y + Z}}} & {y_{o} = {y_{l} = \frac{Y}{X + Y + Z}}}\end{matrix}$

After above steps, the transformation from RGB color space to CIEXYZcolor space to D_(l)x_(l)y_(l) profile connection space is done; nextstep is to transfer value D_(o), x_(o), y_(o) to printer module;

(7) Map D_(o) of scanner profile connection space to printer space: letD_(p)=D_(o) ^(γ) ^(po) ;

(8) Calculate Gamma correction density based on D_(p): D_(p)′=D_(p)^(1/γ) ^(p) ² ;

(9) Plug Gamma correction density D_(p)′ into printer's gray balancepolynomial to calculate gray substitute parameter k_(dd), c_(dd), andcomponent primary which is used as gray core.

c _(p) =c _(dd) =a ₀ +a ₁ D _(p) ′+a ₂ D _(p)′² +a ₃ D _(p)′³ + . . . ,k _(p) =k _(dd) =d ₀ +d ₁ D _(p) ′+d ₂ D _(p)′² +d ₃ D _(p)′³+ . . .

Please note: we only need to calculate k_(dd), c_(dd), or k_(dd), m_(dd)or k_(dd), y_(dd) depends on which type of 4-primary mapping equation ischosen;

(10) Plug k_(dd), c_(dd) into right side of chosen 4-primary mappingequation (type MYK) to get primary value: k=k_(dd), c=c_(dd), m and y;

(11) Convert primary value k, c, m, y into driving value CMYK;

(12) Calculate color X_(p)Y_(p)Z_(p) which is mapped from scanner gamutto printer gamut;

X _(p)=(x _(o) /y _(o))Y _(p) , Y _(p) =Y _(p) , Z _(p)=[(1−x _(o) −y_(o))/y _(o) ]Y _(p)

(13) Map printer (press) gray tone density array [D_(pi)] to gamut ofdisplay device: D_(q)=D_(p) ^(γ) ^(qp)

(14) Calculate Gamma correction value D_(q)′ of density value D_(q):D_(q)′=D_(q) ^(1/γ) ^(q) ²

(15) Substitute variable D_(q) in monitor gray balance polynomial withD_(q)′ to get gray core value c_(dd);

(16) Convert calculated tristimulus value X_(p)Y_(p)Z_(p) intochromaticity coordinates x_(p) and y_(p):

$\begin{matrix}{x_{l} = {x_{p} = \frac{X_{p}}{X_{p} + Y_{p} + Z_{p}}}} & {y_{l} = {y_{p} = \frac{Y_{p}}{X_{p} + Y_{p} + Z_{p}}}}\end{matrix}$

(17) Plug c_(dd), x_(p), y_(p) into monitor's additive mapping equation(type RGK) to get primary r, g, b=b_(dd), and display color'stristimulus value X_(q)Y_(q)Z_(q);

(18) In order to display target color X_(q)Y_(q)Z_(q) on monitor, nextstep is to transform primary value rgb into their driving color space:

R=r^(1/(γ) ^(rd) ^(γ) ^(qw)) , G=g^(1/(γ) ^(gd) ^(γ) ^(qw)) , B=b^(1/(γ)^(bd) ^(γ) ^(qw))

Use RGB as driving value for monitor to display a ready to printpre-proof image.

2. Refer to FIG. 2, the following mapping flow is used to highlight thisinvention's generality, creativity and efficiency features in gamutmapping.

Digital camera predicted color XYZ based on CCD censored RGBvaluetransform predicted color XYZ to Liu's D_(l)x_(l)y_(l) profileconnection space→segment color XYZ to get driving input valueRGB→produce color image on monitor.

(1) Plug R,G,B into color coordinate conversion formula RGB-XYZ toconvert RGB into XYZ;

(2) Plug R,G,B into luminance equation to get luminance value Y;

(3) Plug luminance value Y into gray scale format of Liu's segmentationequation to get gray scale luminance value Y_(gray);

(4) Convert luminance value into gray density valueD_(l)=D_(n)=lg(Y_(w)/Y_(n));

(5) Map D_(n) to monitor gamut density D_(qw)=D_(n)̂γ_(qw); performGamma correction on D_(qw) to get D_(qw)′;

(6) Calculate gray core r_(dd) based on D_(qw)′, then plug gray corer_(dd) into monitor mapping equation (type GBK);

(7) Convert color's tristimulus XYZ into Chromaticity coordinates x_(n)and y_(n):

$\begin{matrix}{x_{n} = {x_{l} = \frac{X}{X + Y + Z}}} & {y_{n} = {y_{l} = \frac{Y}{X + Y + Z}}}\end{matrix}$

(8) Plug x_(n) and y_(n) into monitor mapping equation (type GBK);

(9) Solve monitor mapping equation (type GBK) to get primary value rgband pre-proof display color X_(qw)Y_(qw)Z_(qw);

(10) Convert rgb in primary color space to driving values RGB, then useRGB to ‘drive’ monitor to display color image shot by digital camera onmonitor;

3. Refer to FIG. 3, the following mapping flow is used to highlight thisinvention's generality, creativity and efficiency features in gamutmapping.

Television camera or digital video camera (DV) predicted color XYZ basedon CCD censored RGB value→transform predicted color XYZ to Liu'sD_(l)x_(l)y_(l) profile connection space→segment color XYZ to get TVdriving input value RGB→produce television image on monitor.

(1) Plug R,G,B into color coordinate conversion formula RGB-XYZ toconvert RGB into XYZ;

(2) Plug R,G,B into luminance equation to calculate luminance value Y;

(3) Plug luminance value Y into gray scale format of Liu's segmentationequation to get gray scale luminance value Y_(gray);

(4) Convert luminance value into gray density valueD_(l)=D_(n)=lg(Y_(w)/Y_(n));

(5) Map D_(n) to monitor gamut density D_(qk)=D_(n)̂γ_(qk) and performGamma correction on D_(qk) to get D_(qk)′;

(6) Calculate gray core r_(dd) based on D_(qk)′, then plug gray corer_(dd) in monitor mapping equation (type GBK);

(7) Convert color's tristimulus value XYZ into chromaticity coordinatesx_(n) and y_(n):

$\begin{matrix}{x_{n} = {x_{l} = \frac{X}{X + Y + Z}}} & {y_{n} = {y_{l} = \frac{Y}{X + Y + Z}}}\end{matrix}$

(8) Plug x_(n) and y_(n) into television monitor mapping equation (typeGBK);

(9) Solve above equation to get primary value rgb and pre-proof displaycolor X_(qk)Y_(qk)Z_(qk);

(10) Convert primary value rgb into driving values RGB, then use RGB todrive television monitor, color image shot by television video camerawill be displayed on monitor;

After comparing FIG. 2 and FIG. 3, we can conclude there is no essentialdifference between them except white and black field are swapped.

1. A universal gamut mapping and color management method, and its characteristics are: a. this invention creates a method to do gamut mapping conform to visual effect, thus the mapped color keeps original hue, chromaticity coordinates and luminance which conforms to visual effect; b. to avoid system and random error in color space conversion, this invention adopts calibration target with same structure, perform coordinates conversion in color space and gamut mapping among devices in accordance with same principle and method; c. Primary color's hue is always the same through the whole gamut mapping process flow; d. this invention creates channel primary parameter and reversible power function relationship between channel primary value, reference primary value and driving parameter value; this ensures accuracy of color prediction; driving parameter, reference primary value and channel primary value play important roles in color management system; e. this invention creates a method to generate pure gray scale, and based upon this, it creates gray core parameter, gray balance function and Gamma correction method using density as parameter, thus it ensures image's gray balance and prioritized reproduction of gray tone; f. this invention creates a method to precisely segment color into gray component and secondary color component, and based upon this, it creates Liu's D_(l)x_(l)y_(l) profile connection space; g. this invention creates color space mapping method that can perform gamut mapping and Gamma correction at same time; it can be implemented with analytical algorithm quickly; h. a method to accurately convert primary value derived from Liu's gamut mapping equation to driving value; i. a gamut mapping method among devices; j. in order to achieve this universal gamut mapping and color management method, description of this invention covered a set of Liu's color-matching equations, including Liu's color-matching equation based on subtractive color reproduction, Liu's color-matching equation based on additive color reproduction, Liu's color-matching equation based on 4-primary color reproduction, Liu's scanning color prediction equation and RGB scanning color-segmentation equation, etc; Each equation, formula, concept or phrase that begins with Liu's is creative achievement of this invention, and is part of this claim and its requirements for protection of rights;
 2. A target structure according to claim 1, wherein generalizing cross-media color gamut mapping method, and its characteristics are: color target used to calibrate input and output devices are sample colors generated with same scale level and identical driving value;
 3. a primary hue keeping method according to claim 1, and its characteristics are: a. Liu's primary clamping equation and Liu's reference primary color formula are the fundamental technology behind this method; b. use Liu's primary clamping equation to perform clamping process to the measured value on 3-primary scale in order to get the sample color's clamping luminance value; c. Liu's reference primary formula derived from Liu's primary clamping equation helps to predict primary component's hue and keep hue consistent; the color predicted by the formula shares the same hue as unit primary value; d. the tristimulus value of reference primary always equal to XY_(t)Z; X and Z are sample color's measured tristimulus value; Y_(t) is clamping luminance derived from Liu's primary clamping equation; e. Liu's color appearance keeping coefficient λ is a parameter related to wave length of primary color;
 4. A method according to claim 1, wherein predicting color-matching result accurately using Liu's color-matching equation, and its characteristics are: a. Liu's color prediction equation is the main and key technology behind this method which includes: Liu's color-matching equation based on subtractive color reproduction, Liu's color-matching equation based on additive color reproduction, Liu's 4-primary color-matching equation based on 4-primary reproduction, scan color prediction equation and RGB scan color segmentation equation; b. in Liu's color-matching equation, every primary color has its own channel primary parameter in each channel; there exists reversible power function relationship between channel primary parameter and reference primary color; the same goes with relationship between reference primary color and driving parameter; c. Liu's color-matching equation can predict primary synthesis result accurately; d. RGB scan color separation equation and Liu's scan color prediction equation are twin equations describing the same sample color;
 5. A method according to claim 1, wherein characterizing Liu's color-matching equation and its characteristics are: a. use Liu's primary clamping equation and Liu's reference primary formula to calibrate sample color on three primaries scale and give primary color independent color-matching feature; b. calibrate primary color's three channel primary value using primary scale's reference primary series to ensure space independency of each primary in 3 dimension color-matching space; c. this method establishes power function relationship between reference primary parameter and color driving value;
 6. A method according to claim 1, wherein creating pure gray scale for scanner, printer, normally-white-monitor and normally-black-monitor, and its characteristics are: a. use the pure gray scale created with this invention as basic model of gray tone reproduction; b. the source data for creating pure gray scale are initial luminance array [Y_(oai)], [G_(oai)], [Y_(pai)], [Y_(qwai)], [Y_(qkai)] measured on color scale displayed on scanner, printer and monitor; based on these source data, the ideal luminance arrays of pure gray scale without optical distortion are created after nine conversion steps; the optical distortion is eliminated using special power function fitting method; c. on pure gray scale, chromaticity coordinates of every dot is identical to that of media white dot;
 7. A method according to claim 1, wherein characterizing pure gray scale component primary value; and its characteristics are: a. pure gray scale component primary value is the result of calibrating tristimulus array of pure gray scale using Liu's color-matching equation; b. pure gray scale component primary value is power function with pure gray density as independent variable; for 4-primary printing, it is polynomial function with pure gray density as independent variable; c. the minimum value in pure gray scale component primary is considered the gray core; d. Gray core is used as a tool for prioritizing gray tone reproduction, a carrier for Gamma correction and key to fast analytical operations; e. in pure gray scale component primary polynomial function created for 4-primary printer device, black primary value is purified reference primary value; f. let k_(dd)=Q(d_(k)̂γ_(k))^(n), in which k_(dd) represents gray component substitute value; g. the 4-primary gray balance polynomial calibration method is: pre-define black primary color's tone curve to convert 4-primary color-matching equation into 3-primary color-matching equation, then perform calibration on 3 primaries gray balance polynomial using 3-primaries color-matching equation;
 8. A method according to claim 1, wherein performing Gamma correction on image gray tone, and its characteristics are: a. pure gray density of pure gray scale is used as Gamma correction parameter; b. Gamma value equals to 2; c. cmyk 4-primary reproduction is done by substituting pure gray density parameter D in gray balance polynomial with Gamma correction density D; in other words, it is accomplished by using Liu's 4-primary printing anti-Gamma gray balance polynomial;
 9. A method according to claim 1, wherein creating D_(l)x_(l)y_(l) profile connection color space for scanner and its characteristics are: a. Liu's color segmentation equation and its gray scale format are important tool and essential technology to create D_(l)x_(l)y_(l) profile connection color space; b. perform color segmentation on scanned color XYZ using Liu's color segmentation equation to get gray color p, XYZ is derived from Liu's scan color prediction equation; c. this method performs color clamping on gray component using parameter p and on color component using parameter (1−p); d. density parameter D_(l) of D_(l)x_(l)y_(l) profile connection color space depends on parameter p; density parameter D_(l) is calculated using gray scale format of Liu's color segmentation equation; chromaticity parameter x_(l)y_(l) depends on segmentation color XYZ; e. Liu's scan color segmentation equation has 3 sub-types; it divides the color to-be-segmented into 3 areas; f. when transforming or compressing color XYZ to target color space, D_(l)x_(l)y_(l) color space has high compression ratio, constant luminance and chromaticity; g. Liu's color segmentation equation can be simplified to ternary quadratic equation for fast operation;
 10. A method according to claim 1, wherein calculating twin primary value c′m′y′ for scanner in RGB color space with Liu's 3-primary clamping equation, and its characteristics are: a. Liu's 3-primary clamping equation introduced in this invention is an efficient tool and core technology for calculating twin primary value c′m′y′; b. Liu's 3-primary clamping equation performs clamping on twin primary value c′m′y′ using color appearance keeping parameter λ and gray core parameter to ensure c′m′y′ possessing reference primary characteristics, meanwhile the hue and chromaticity coordinates of color RGB stay unchanged; c. Liu's 3-primary clamping equation is ternary quadratic equation and can be solved quickly using analytic method;
 11. A method according to claim 1, wherein converting scanned color from RGB color space to CIEXYZ color space, and its characteristics are: a. it is a conversion method in accordance with chromaticity theory instead of an approximation methods based on regression analysis theory; b. calculating gray core parameter with Liu's color segmentation equation ensures calculation accuracy of gray component in color RGB; c. Liu's 3-primary clamping equation ensures the hue independency of twin primary value c′m′y′; d. c′m′y′-cmy twin primary conversion expression mentioned in this invention helps to convert color coordinate accurately; e. RGB to XYZ conversion is carried out using scan color prediction equation mentioned in this invention; f. RGB-XYZ coordinate conversion is carried out using RGB scan color segmentation equation and Liu's scan color predication equation;
 12. A method according to claim 1, wherein creating D_(l)x_(l)y_(l) profile connection color space for digital camera or digital television camera, and its characteristics are: a. calculate tristimulus value of gray tone on standard monitor using gray scale equation in Liu's color segmentation equation; the clamping parameter Y in this equation is luminance value derived from luminance equation; b. calculate gray tone luminance Y_(grey) using luminance equation in Liu's gray scale equation based on luminance value Y of color XYZ, then convert Y_(grey) to density parameter D_(l) in color connection space; c. calculate x_(l), y_(l) of D_(l)x_(l)y_(l) profile connection color space using XYZ derived from RGB-XYZ matrix equation; d. the method to create D_(l)x_(l)y_(l) profile connection color space can be universally applied to various types of display devices such as CRT, PDP, LCD, LED, etc;
 13. A method according to claim 1, wherein mapping parameter D_(l) in profile connection space to gamut of target device, and its characteristics are: a. establish power function relationship between profile connection space parameter D_(l) and pure gray tone density of target device gamut to achieve gray tone mapping objective; this method can be universally applied to various types of output devices; b. the method of performing gray tone mapping using pure gray density parameter is not only applicable to different type of devices, but also applicable to devices producing color based on additive or subtractive method;
 14. A method according to claim 1, wherein mapping color D_(l)x_(l)y_(l) in profile connection space to gamut of target device, and its characteristics are: a. Liu's gamut mapping equation is the core technology to implement this method; b. color D_(l)x_(l)y_(l) in profile connection space is provided to Liu's gamut mapping equation on two streams and intersect in Liu's gamut mapping equation; c. the channels transferring D_(l) perform Gamma correction on gray tone, then calculate gray core parameter based on Gamma correction density, finally plug gray core value dynamically into Liu's gamut mapping equation; d. plug chromaticity parameter x_(l)y_(l) into Liu's gamut mapping equation, along with Liu's clamping luminance Y_(p), Y_(qw), Y_(qk) and gray core value to perform clamping on Liu's gamut mapping equation; e. Liu's gamut mapping equation is a color segmentation engine; the reference primary values derived from it all inherit hue constant characteristics; f. primary driving input value can be calculated based on reference primary value derived from Liu's gamut mapping equation; g. Liu's gamut mapping equation is ternary quadratic equation and can be solved using analytical algorithm;
 15. A method according to claim 1, wherein mapping color XYZ captured by scanner to gamut of printing device; it inherits all the general characteristics of Liu's gamut mapping method and also has the following characteristics: a. gamut mapping is carried out with the help of Liu's scanner-printer mapping equation; b. this method calculates component primary value of gray tone and gray substitute parameter k_(dd) using Liu's 4-primary printing anti-Gamma gray balance polynomial Gray component primary value is used to calculate gray tone Gray component parameter k_(dd) is a Gamma corrected gray component substitute parameter with pure tone distribution curve; c. the color X_(p)Y_(p)Z_(p) produced after mapping is the predicted color to be printed; d. the driving value CMYK which produces color X_(p)Y_(p)Z_(p) is calculated based on reference primary cmy and k_(dd) derived from Liu's scanner-printer mapping equation; CMYK is represented using double Gamma correction function;
 16. A method according to claim 1, wherein mapping color XYZ captured by digital camera to gamut of normally-white-monitor; it inherits all the general characteristics of Liu's gamut mapping method and has the following characteristics: a. gamut mapping is carried out with the help of Liu's digital camera-monitor mapping equation; b. Liu's digital camera-monitor mapping equation has three sub types With join effort of Y_(qw) and r_(dd), Y_(qw) and g_(dd), Y_(qw) and b_(dd), the method ensures the color mapped to target gamut can inherit its hue and chromaticity in source gamut, and the luminance distribution of both colors has decent similarity; c. the color X_(qw) Y_(qw) Z_(qw) produced after mapping is the predicted color on normally-white-monitor; d. the driving input value RGB of color X_(qw) Y_(qw) Z_(qw) produced by normally-white-monitor is calculated based on reference primary rgb derived from Liu's digital camera-monitor mapping equation RGB is represented using double Gamma correction function;
 17. A method according to claim 1, wherein mapping color XYZ captured by television camera or digital video camera to gamut of normally-black-monitor; it inherits all the general characteristics of Liu's gamut mapping method and has the following characteristics: a. gamut mapping is carried out with the help of Liu's television camera—TV monitor mapping equation; b. Liu's television camera—TV monitor mapping equation has 3 sub-types; with join effort of Y_(qk) and r_(dd), Y_(qk) and g_(dd), Y_(qk) and b_(dd), the method ensures the color mapped to target gamut inherits its hue and chromaticity in source gamut, also the luminance distribution of both colors has decent similarity; c. the color X_(qk) Y_(qk) Z_(qk) produced after mapping is the predicted color on normally-black-monitor; d. the driving input value RGB of color X_(qk) Y_(qk) Z_(qk) produced by normally-black-monitor is calculated based on reference primary rgb derived from Liu's television camera—TV monitor mapping equation RGB is represented using double Gamma correction function;
 18. A method according to claim 1, wherein mapping color XYZ generated by printer device to gamut of normally-white-monitor; it inherits all the general characteristics of Liu's gamut mapping method and also has the following characteristics: a. gamut mapping is carried out with the help of Liu's printer—normally-white-monitor mapping equation; b. Liu's printer—normally-white-monitor mapping equation has 3 sub-types; the method ensures the color mapped to target gamut can inherit its hue and chromaticity in source gamut, and the luminance distribution of both colors has decent similarity; c. the mapped color X_(qk)Y_(qk)Z_(qk) is the preview of color on normally-white-monitor; d. the driving input value RGB of color X_(qw)Y_(qw)Z_(qw) generated by normally-white-monitor is calculated based on reference primary rgb derived from Liu's printer—normally-white-monitor mapping equation; RGB is represented using double Gamma correction function;
 19. A double Gamma correction method according to claim 1, wherein converting device's reference primary value to driving input value; its characteristics are: a. this method can be universally applied to devices producing color based on either additive or subtractive color theory; b. calculating input driving value based on reference primary value generated with mapping method is an anti-Gamma tone correction method with dual correction feature; it performs anti-Gamma correction on both primary tone itself and pure gray tone. 